Okay, so it sounds like she is biting for different reasons. Biting when you take something fun away is "resource guarding". When you pick her up to prevent her from tangling with other dogs, she is probably redirecting her fear and aggression onto you.
There are some established methods to reduce resource guarding. You can google this, or search YT for this term. I think there are some other active threads going on about how to fix this behavior.
The fear-based aggression - it's all about distract and reward. I know it's difficult. I never allow Thor in front of me on walks, and even after months and months of practice, he STILL runs ahead of me to growl at other dogs. It takes a lot of patience and persistence.
I would work on the resource guarding first, because this behavior is easier to fix. The work you do with her there might spill over into her overall demeanor. The key to both is to catch her before she's worked herself into a frenzy. Easier said than done, I know.
I would also recommend more exercise. Whatever she gets now, add 15-30 minutes more per day of a strenuous activity, like fetch or keep away.
As challenging as it is, I urge you to work hard on this, because it's not really acceptable for her to bite you. I think this probably does indicate a problem with the chain of command in your house. That might also be exacerbating her aggression problems - her belief that she's in charge may be causing her a lot of anxiety, which keys her up to bite more.
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