I think in your case you need a good rack. At least that is what we use for indoor drying. We have this one:
http://www.ikea.com/us/en/catalog/products/50095091/. (I experimented and it fits a whole load). And your roommate might just have to get used to seeing your undies, or if you really really really can't get past that, put JUST your undies in the closet, but yes, with air between them.
If I understand correctly, your shirts/pants ultimately hang on hangers? I used to do mine that way, but you will have to experiment a little. Some shirts won't like to hang normally on a hanger when wet (they will stretch out of shape) and some will be fine. I had the best luck with pants in those sandwich-type hangers and hanging them upside down.
If you stick with the hanger plan, I am envisioning something for your balcony - perhaps you could zip-tie two poles kitty-corner from each other onto the balcony rail, but going further up. Then attach a rod of some kind between them. Then at least in nice weather you could hang the clothing outside. Or! Or! You could get one of those rolling closet-rack things at a big box and hang the clothes on it if they will be on hangers, but out in a bedroom or other open area, not in a closet.
You don't need a TON of room between the clothes, but they can't hang just as they might in your closet. There does need to be air circulating among them. Similarly, it wouldn't be a good idea to, say, fold your pants in half and hang them over the hanger. They'd never dry.
Give it a shot. I think you'll be pleasantly surprised at what you come up with! It's pretty easy, and if some method is not working, invent another one . . .