Won't hold a charge off the built-in charger/convertor? If it's an older TT, the built-in non-regulated transformer charger/convertors never charged your battery fully anyway!
Take it to a battery shop and have them test it so you know for sure if it's dead or not.
Your battery needs are completely dependent on your usage. Planning to boondock a lot? A lot of your battery draw will be the furnace fan overnight. Add to that interior lights, radio, some fridge usage...
The biggest benefit of a sealed/Optima type AGM (Absorbed Glass Mat) is that you can mount them in strange positions, on their sides, etc. without concern of spilling.
"Flooded" batteries hold more energy for less dollars. If you don't "need" an Optima type battery, skip it.
A lot of people convert to a pair of 6V golf cart batteries in series = 12V. This may or may not be more than you need. Check the local golf cart shop to see if they've got a pair of used batteries (golf carts run on six 6v batteries in series to get 36V, and often only one or two batteries will go bad, leaving the other 4-5 without a "match" to go out the door with a customer). I've bought them for $40-50/ea and they usually have a 5-7 year life span if they're taken care of.
Change out your old convertor/charger for a newer smart charger or buy an automotive smart charger.