Stems come in a wide range of lengths (45-130mm), and they are generally fairly cheap (starting around $20, and generally around $50). It is very easy to remove the existing stem and install a different one. So I think it is worthwhile to get the shortest reasonable-looking stem you can find and give that a try.
Raising the handlebars is not so easy, other than by getting a different stem. Unless there are spacers above your stem on the steerer (the part of the fork that the stem is attached to), in which case it is extremely easy to remove the stem, move the spacer below the stem, and reinstall the stem. It is fairly rare that a bike is sold with extra spacers above the stem. Beware: Sometimes there is a small (5mm) spacer above the stem that has to stay there; its purpose is to prevent the steerer cap from touching the steerer.
Another way of raising the handlebars would be getting different handlebars with more rise built into them -- generally the bars would be horizontal in the center, then bend up at an angle, and then flatten out to horizontal at the grips. I don't think this would help much, because the amount of rise you get out of these bars is usually not very large, maybe an inch.
Here is something else to consider: I'm 6'5", and for years I was riding a mountain bike that was too small for me, because it was the biggest frame that I could easily find in my price range back when I bough the bike. I got quite used to this bike, so when I finally got a new bike with a larger frame, I felt quite stretched out when reaching for then handlebars. Eventually I got used to that and I doesn't cause any problems even on long rides. But if you've had this bike for several years and it still bugs you, then you should probably get a short stem and see if that makes it feel better.