I believe this is quite common behavior for bunnies. Unfortunately, and against popular belief… most bunnies DO NOT like to be picked up. There are always a few exceptions to that as well as a varying scale of rabbits who tolerate it to different degrees – but I have found that rabbits tend to be born with a certain personality and it most often sticks with the bunny their whole life. Part of this personality is how much they care to be held/cuddled.
If your buns are the types that do not like to be picked up – just your presence in the room will likely create anxiety in them because they know you are likely to go and pick them up… and that makes them nervous and unable to relax and be happy. I had a bunny who HATED to be picked up, yet we were still able to have a very loving and trusting 10 year relationship. Despite having handled her for 10 years worth of nails clippings/fur brushing/medical necessities and vet trips… she hated to be picked up just as much when she was 10 than when she was a baby. I simply never picked her up unless I HAD to. I mostly laid down/sat down on the floor with her when we interacted … played/gave her bunny rubs/let her explore and hop around me/gave her treats. I never forced the interaction and she came to totally trust I wouldn’t do anything to her she didn’t care for.
My best recommendation for you is that you try to see how they begin to behave if you stop picking them up whenever you see them…… and try to spend time with them that includes floor time play where you are not trying to handle them. Instead… try to devise a way to set their cage up so that they can get in/out of their cage on their own. Getting them out is usually the easy part…. getting them to go back in on their own is going to be the challenge (because they don’t usually WANT to go back in!). But do your best and try to be creative and see what you can come up with. We used to gently “herd” one of our bunnies back into his cage (adding a tiny banana reward to the cage when it was time to go back in also helped) when he had one. When he realized his option was either to go back in on his own or to eventually get picked up… he quickly learned to pick the option he liked best and began to go in on his own when we wanted him to. It might take a bit of time for them to realize they won’t be picked up and begin to relax around you…. but I will bet you will begin to see the real personalities of your bunnies if you do. Good Luck!