Some turtles just really don't appreciate human interaction and have more "wild" instincts than others.
For example, my one RES can't really stand me, while the other is more docile and I can pretty easily hold him. I just let the aloof one do her thing and accept that she's not really a touchable pet.
As for your situation, try covering the sides of the tank where the turtle basks. She may appreciate the privacy. It's perfectly normal for RES to be shy baskers and anything to help her get used to a little background noise is a step in the right direction.
For the biting, there's not really anything you can do about that. But, turtles are really pets to be looked at and not touched, so not handling your turtle really shouldn't affect your life too much. Be sure to let her see you feeding her. Once turtles associate humans with food they are always more "friendly" towards them and they'll start begging and looking forward to seeing you.
When you have to handle your turtle, just do so carefully. Be sure to support the shell on the top and bottom and just keep an eye on where your turtle's head is. My one turtle likes to bite, but she's pretty clear about her intentions and when I see her sticking her head out and opening her mouth I just make sure my hand isn't near her

. Turtle bites are pretty painful so I can understand why you're frustrated.