There are various ways that handles are attached and without seeing them or at least a description of them, I don't think anyone will be able to help you. It may be that they have corroded and seized in place or, it may be that they actually screw in (not as you described but where you would actually screw the entire handle and the extra screw would be so you can adjust the marking to where you could read it (hot & cold). Also, the screw you removed could be part of the valve. I was working on a kitchen sink once and took out a screw that appeared to hold the handle on but it was actually part of the mixture/flow valve and the water pressure pushed it all out. I had water everywhere. Be careful and take precautions until you are sure off what you are doing.
The faucet 'seals' itself to the unit because the minerals in hard water solidify on the splines where the faucet handle contact the base part of the valve. These minerals, over time act like rust, or a hardness that seals the exterior handle. Gently pry around the perimeter, of the back of the handle part and the wall. Eventually this will break the hardness of the minerals, and allow the handle to be free.
Get a plumber as it sounds like the inside washer has shot - it is quicker and safer than mucking around - buy the right new taps beforehand or the plumber will do you extra for any he has to buy - go to a cheap DIY place and get innards and handles, just in case. Some plumbers charge twice for doing two! But make sure that get both done at same time, then you won't forget which needs doing next, especially if you're doing only the cold one now. If you use pliers to pull up the ring washer it releases water out of the tap. You could be waiting ages for a bath and then have a drippy tap!
With a hammer lol no you use a screw driver 2 get it off then install the new 1
Try pulling the handles directly straight out. Or you may have to unscrew the whole line where they are connected to the hot and cold lines. Be careful not to strip it else you'll be stuck. Good luck
If your gonna do it, take some large water pump players and grab the whole handle, gently begin rocking it sideways and around carefully as to not bend the valve stems they are connected to.