Some categories are unavailable when making a new post

After upgrading to a new server my WordPress site is experiencing issues with categories. All of my categories are still there, I just can’t use them for new posts. I can see them all when I visit Posts > Categories within the admin, and they are still visible on the front end of the site along with the posts that were previously made. However when I try to make a new post only about 5% of the categories are available.

Nothing should have changed with the site, it was directly copied from one server to another by my hosting company. I have contacted them about this issue and I am waiting to hear back, but in the meantime is there anything I can try to fix this myself?

I’ve done a lot of searching and I can’t find anyone else who has experienced this so I have no idea where to go with this problem.

Thanks in advance for any suggestions!