First off, no it is not possible to physically replace the 4 speed transmission in your car with the later 6 speed TF-80SC.
The reason Volvo needed to use the GM 4T65E transmission behind the in-line 6 engine is simply packaging - the motor is so long (ie wide when installed in the XC90) thet there's no room at the end of it for a transmission. The 4T65E uses a chain drive on the engine end and locates the actual transmission behind the engine block.
Secondly, the transmission is electronically controlled, and is integrated via canbus to numerous other modules to work everything from transmission operation to stability control and ABS...the computer system wouldn't know how to control a different transmission.
You could spend the $6000 on a transmission rebuild which may sort out the transmission issue (and these days there are upgraded kits and plenty of specialists who have rebuilt these units !) but the XC90 is a very complex car - there will absolutely be suspension work, steering angle sensors, power steering pumps, turbochargers, PCV systems, fuel pumps...a real multitude of potential (let's face it, LIKELY) other failures which are going to pop up in the future. The money you spend on getting it to work again is ONLY going to restore it to the condition you bought it in - a $3100 old Volvo.
I appreciate WHY you bought your car, but unfortunately there's no way to run an old XC90 (especially the turbo 6) on a limited budget.
If you are on a budget, I'd highly recommend pulling the pin before you're over-committed to that particular vehicle. (ie from a financial position, it would make more sense to spend $6000 on a 2.5T version with a much more reliable (but still nowhere near perfect) AW55/51SN transmission... this is always the "go-to" budget-conscious XC90 choice.
The reason the T6 models are so cheap is precisely because their owners are in exactly the same position as yourself. I know of one family who bought another car for daily uses and only use their T6 for the short commute to the train station - faulty transmission and all. The cost of a transmission rebuild doesn't make any economic sense on a cheap, 16 year old car of any brand.
Yes the used market (particularly AWD wagons) is stupid in Canberra or anywhere else at the moment, but there are options. For example, the 1999-2003 Subaru Outback models are quite safe in a collision, don't have Takata airbags like the later model and are common in wrecking yards (meaning used transmissions are cheap and easy). Mechanically they are robust and will last a very long time, their cabins are usually long enough to sleep in and they have capable AWD systems. The only real thing of note is that the 2.5 base engine (still the one to look for) is known for leaking oil from the head gaskets - factor in some maintenance (having the heads rebuilt and using Turbo gaskets). Something like this: https://www.gumtree.com.au/s-ad/blacktown/cars-vans-utes/subaru-outback/1275022626
You will have no problem getting it correctly serviced at any garage, and it's very unlikely to break the bank. (If it does, you can just dump it and buy another one...)