she is new to you. she sounds insecure and may not know what she is supposed to do. i would go to the woods a lot and try to kick up rabbits. i would also get some rabbit scent and make some sent trails. also maybe a starting pen or places where there are rabbits and get her to chase the rabbits. once the light goes on in her head she will know more what to do. these are some things i have done with my beagle puppy. i am not an expert i am new but these things have worked for me. i think your dog likes you and trusts you so it stays by your side right now.
i would not leave her at a pen with someone for an extended period of time like two or three weeks. i have heard some horror stories. i would go with the dog to the pen and be a part of the training. i have a couple of places in town where i live that is infested with rabbits. i go at night when there is no traffic or anyone around. i take a flashlight and the dog sees the rabbits and sets chase. she also picks up the scent trails. this has really helped in the woods. she is really focused on finding a rabbit.