You are thinking logically which is good - too many rush out and buy new without understanding what they need and want long term. Experiment cheaply for a bit and learn what appeals before investing.
As that goes, saddles move from bike to bike so they are an good area to invest in first, as are pedals. Even tires to some degree, if you stay in the same wheel size, can travel bike to bike.
The stem issue can be easy or not, depending on how handy you are and if you can look for bargains. Certainly a shorter stem is possible, it's one of the first things I change on those types of mtbs because the stock ones (as in the pics of a 93 GF online) are very aggressive in terms of reach. Shorter and more of a rise. If you look at my stem you see it's actually quite a steep angle.
You can either look for a quill/stem combo like I think your bike has (but shorter) or swap that out for a threadless conversion quill and a modern stem. The problem is then finding a modern stem with the old school diameter for your current bars. Otherwise you start to need to also swap out bars, brake and shifters to match the new diameter stem. They are out there, you just have to look and want to convert/adopt components (hence the handy comment). If you take it to a shop, unless it's a family/budget business, they'll just say it can't be done and tell you to buy a new bike.
If you want to do the work on the stem then ride what you have for a bit and see what appeals to you. Look for a comfortable seat. If the goal is light/credit card touring a more road specific bike might be the ticket and if you can work on them, used bikes can be had at reasonable rates.
I have transitioned out of that mtb into my current road bike which, because I bought a last years discounted model, netted me a new Aluminum and Carbon Fiber disc brake Taigra endurance bike that retailed $1500 for $800. But, I would not have known what I wanted out of it if I hadn't spent time actually touring first. WHen I took up touring again I thought the answer was a Surly LHT at a minimum of $1500 used (local rates) which, while good, is now what I don't want in a fast touring bike.
Fully loaded
Light tour
Afternoon ramble
