Will a hammer drill also fix screws?

What you're after is usually known as a combi drill. It's a combination of an ordinary drill and a hammer drill.

With regard to the "weak wrists" aspect, I suggest wearing padded gloves (like cycling gloves) because many drills, when stopped, give a sudden kick-back torque which is more of a problem than the weight of the drill. It makes the difference between pain the next day and no pain.

An impact driver, which you mentioned, is for driving screws and nuts at a fairly high torque (think of the tool a garage mechanic uses to remove wheel nuts). An impact driver is not what you need for the described tasks in order to keep in budget.

This isn't a site where product recommendations are on-topic, but the Bosch PSB 1800 is a competent tool and fits in your price range. The same manufacturer does a drill accessory set which is likely to include all the bits you'll need for what you describe.

Do have a practise go on something when you get a new tool just to get an idea of what it does.

If you find that a hole isn't being drilled at a reasonable speed, poke into it with a screwdriver to feel if there's some obstruction of a different material than you're expecting: if you're using a masonry drill bit then it won't go through metal, and the metal could be a water pipe or conduit for electrical cables.

If you are buying screws for putting things up, Torx screws are easier on the wrist because you don't have to push on them for the driver bit to stay engaged.

For assembling flat-pack furniture, it's best to use hand tools for the first couple of fasteners to get a feel for how they work - that's what they're designed for - and see if you think a power tool would actually be any better. (I would however suggest investing in a set of Wera metric hex-plus keys if you have a lot of furniture to assemble which uses hex socket screws. And a wooden mallet.)