Does my battery need to be replaced?

Batteries deteriorate "due to time". The time clock starts the minute the acid is poured into the lead plates, NOT the day the battery was put into service. Six years of service, is a "very long time" for the life of any lead/acid battery.

The reason your battery will not take a charge is because it has developed sulfate coating on the lead plates, effectively "insulating" them from the electricity. It needs to be replaced.

You can extend the life of any battery dramatically by keeping it at 100% charge, "all the time", as it will discharge on it's own slightly each day. You want to keep the new battery plugged into a Battery Maintainer 24/7. A Battery Maintainer, is "a regulated type" of a battery charger and it will not cook a battery when it's connected 24/7.

A 2.5MA size maintainer is plenty to keep your battery a full charge. They are about $38 to $65 depending on if you buy a brand name.

Also you have to maintain by making sure you battery liquid is always covering the plates by pulling the caps and adding distilled water up to the bottom of the fill neck for each cell.
The moment you don't do this the coatings on the plates (mentioned by John Palmer above), will dry, flake off and fall to the bottom of the battery. Enough of this debris on the bottom will short out the plates, and the battery fails.
The only ones you don't have to check are the marine "Gel" batteries. Even normal auto batteries that are supposedly maintenance free will need to pull the flat caps off and check every so often again, topping off with distilled water only not tap water or drinking water.
Also keep in mind too, when you buy a battery now a days you need to check this liquid (acid) level as soon as you get it home. I have been getting batteries lately that have only been filled 3/4 of the way full. I think they do that for 2 reasons. One being spillage during shipping or the person filling with acid the first time at the store only uses 1 container of acid for each battery instead of like 1 1/2 container like they usually need. It's cheaper for them not to fill all the way and will also make your battery fail right after the warranty is up like 2-3 years, so you'll have to come back sooner for another battery.
By the way, I have never heard of a battery lasting 6 years unless you regularly maintain as per these instructions and even then it's rare.

It might not be dead, just discharged from sitting too long with no charger on it. But it's probably lost some capacity. Check the water level and leave it on the charger for a couple of days and see where it gets you.

You can get a hydrometer at an auto parts store for less than ten bucks. It's the only accurate way to assess both state of charge, and battery condition (by making sure you've actually got a full charge, and comparing each cell to spot weak ones, before it fails on a trip). If it's been charging for days and the hydrometer still shows it being low, it's probably close to death. If the specific gravity of one cell is much lower than the rest, there's your culprit.
www.google.com/search?q=hydrometer&oq=hydrometer&aqs=chrome..69i57j69i60&sourceid=chrome&es_sm=93&ie=UTF-8#tbm=shop&q=hydrometer+for+car+batteries&spd=7871197011028455301

You should never let your battery drain below 50% (if you want maximum lifespan), because it's during deep discharges that you get the most sulfation on the plates, and each time you go below 50% you can damage it a little more. And recharging doesn't convert all the sulfation back into lead and acid, so it builds up on the plates as time goes by, making each future charge less effective, and so on and so on until it dies. A battery only has so many deep discharges in it before it refuses to take a full charge any more. So that 20% discharge you start out with is 40% of your intended usable capacity gone before you've even left the house... You've probably been lessening your battery capacity, even if only slightly, every time you go out at only 80%, and maybe now it's catching up with you.

As an aside, the LEDs on RV converters don't know squat about your battery. Your typical RV converter can only get your battery to a REAL full charge if it's left on all the time, because almost all of them charge at too low a voltage. They charge quickly up to 80 or 90%, then the amps start dropping off and it takes days, maybe even weeks, to get that last 10-20%, even though the pretty green light is on. Trojan recommends charging unsealed batteries at 14.8 volts, and no converter I know of will do that. Even my Progressive Dynamics "Charge Wizard" only puts out 14.4, in short-term boost mode, to 90% charge, then it drops to 13.6, and then down to 13.2! That takes forever to actually get the battery to 100%, even if it isn't noticeable during use.

My Morningstar solar charger, on the other hand, lets me choose from many voltage schemes, including 14.8. I haven't run my generator for anything other than power tools in over a year (I'm parked off-grid, so it's solar or generator power only, no plug-in charging). And that's using the original incandescent lights, a hot air furnace at temps as low as 14f, my laptop, desktop with 75 watt surround sound, a 26-inch tube TV (since replaced with a 24" flatscreen), boom box, shop vac, drill, a little microwave action, Christmas lights... I know that every time I go out there the battery will be FULLY charged.

Sealed batteries take even longer to charge because they can't handle as much voltage as "wet" batteries without gassing too much and blowing the safety valves. They're designed that way, it's the only way you can make a sealed battery. I'd never use a sealed battery myself, they're too expensive, have too little capacity per pound, and they're too slow-charging because of the low voltage requirement. Maybe if I was rich enough not to worry about battery costs, but probably not even then. I'm not into "new" technology if it costs more but does less.

Car batteries are not great camper batteries, and neither are the so-called "marine batteries". They're both engine-starting batteries, built with thin plates with lots of surface area, so they can throw all their amps at the starter at once. They're not intended for deep slow discharges like you get camping. Real deep cycle batteries have heavy plates that resist sulfation and deterioration much better. They're built for slow, deep discharges and a long life. Before I bought my current 6v golf cart batteries, I was using two big 6v Deka solar batteries that had been installed in an island cottage for at least 10 years, AND were dropped off a forklift, before I got 'em. I used them for three years, and I could still be using them, but they were too big and heavy at 115 lbs each, as well as having broken seals around the terminals, from the plates shifting when they were dropped. I'm sure they had lost much of their original capacity as well, but they were so big that it didn't matter for my usage. I replaced them with two Duracell 6v GC batteries from Sam's Club, at half the weight of the old ones, for about $230. I love 'em.

For towing, buy a battery isolator switch, also at the auto parts store and matched to the amperage of your alternator (or higher), and run wires back to the camper battery (best option is using bigger wires than those in the tow harness to prevent the voltage drop of small wires robbing half the charging power before it gets to the battery). It mounts in your vehicle, and after the alternator charges up the car battery, it switches to charging the camper battery. That way you can leave it connected while you drive. If you just hook both batteries to the alternator, and they're not identical, then the bigger one will always be overcharged and the smaller one will always be undercharged.

Good luck!