Hello!
I've recently inherited a banana plant, and it is not doing well. Its very wilty, some of the old leaves were so soggy they were mush. One family member suspected a fungus. It has some gross looking black marks on it.
I think its been over watered (too frequently) as well. Some of the older leaves were practically mush. Should I suspect root rot/overwatering? Do any of the weird spots look like a fungus? How do I treat those issues?

The leaves are very fragile right now, just moving the plant can cause them to break. Theres only two that arent broken, and one new leaf growing in. Should we remove the leaves that are broken?

The picture is of the plant outside, but it's inside now. It was only out for a day or two, although it got pretty cold one night. It was having issues before that though, which is why they put it outside.

Its leaning pretty heavily to one side, we are thinking about tying it to a pole to keep it upright.

UPDATE: Wait, I know this is the opposite of what I just said but after some searching we think its probably under watered. We were only watering this once or twice a week for the past few weeks. And wasn't watered for a week before that. Is resuming a healthy watering schedule going to perk it back up? Anything special we need to do?

Please, any advice on how to save this plant is greatly appreciated. It means a lot to us.

Ensete are banana relatives but not bananas. Their care is pretty much the same. The major difference between Ensete and Musa are that Ensete rarely, if ever, pup.
It does look underwatered.
It needs a position where it can be protected from the wind and extremely bright hot full sun, wind will shred it. The sun can burn it unless it's been acclimated to full sun. They do also like high humidity and can look ratty until the rains start up and humidity increases in the spring into summer.
It appreciates food, a sprinkle of a slow release or regular feeding with water soluble.
Old leaves also do fall off, old being the lowest on the stalk. That is how the plant makes a trunk.
I have 5. Four in containers, one in the ground.

Oh, okay. Thank you! Do I need to be concerned about those black spots on it? (Also, your plant looks very nice!)

Probably not yet. Its been winter, even in a greenhouse and outside, adverse things happen in winter