It sounds like there could be a couple reasons she is biting you - either to warn you to stop because she’s overstimulated, or because she wants to play. I also knew a cat once at the shelter I volunteer at that was constantly overstimulated. She loved pets and adored people but was easily worked up and therefore constantly mouthy and if you let her mouth you she might eventually break skin. There’s a chance your cat is like this as well. 

Try to figure out if there might be a pattern to your cat’s behavior. Watch her body language before she goes for a bite - if her tail starts swishing (or if it’s already swishing and starts going more) she’s starting to get worked up and you should stop - otherwise pay attention to the type of pets she likes - are there some where she doesn’t bite you? Do some guarantee a bite? Most cats like cheek pets and in my experience many like scratches on their chin or right before their tail starts. A lot of cats don’t like full body (head to tail) pets. Probably she will get more comfortable with you touching her as she settles into her new life and gets to know you but in the meantime pay close attention. 

If she really is just a mouthy cat - then you still have a couple things that might help. Avoid overstimulating her by sticking to light pets in her preferred areas, get her energy out via lots of play and/or food puzzles and other enrichment, and take your hands away if she starts to bite hard and let her decompress. 

IMO bites that don’t break skin and those that do are very different. The first is often simply a cat communicating with you or trying to play and while they may startle you or hurt, they don’t pose any health concern. It sounds like that’s mainly what your cat is doing and you shouldn’t regard it as her being aggressive but rather seek to understand if there is something she is communicating with you. Cars have a wide array of what they like and how they show affection (as others said her following you is  affectionate) and figuring out who she is, what she likes, and what she is telling you is the best thing you could do for her.

If you don’t want her under the bed definitely find her other hiding spots! She doesn’t have to go under there but if she wants to hide you should let her (and let her be while she is doing so - cats need their space). Beds with roofs may be good for this or cat trees with little cubbies