I have a shared version of the player's handbook but I can't choose a forest gnome as my character's race.  I have tried rebooting, deleting and reinstalling the app, and had my DM drop me from the campaign then add me back; still can't choose.

Is the character in question in the account in which the content is being shared? See, the content isn’t being shared “with you,” it’s being shared with that campaign. If the character is in the campaign, then please see this helpful FAQ for useful suggestions: (https://www.dndbeyond.com/forums/d-d-beyond-general/bugs-support/142100-how-to-access-content-on-d-d-beyond-faq).