my apps aren't updating and it says that it is because I downloaded them with a different Apple ID, but isn't there some way that I can update them with my new Apple ID?

Those purchases are tied to the other ID and can't be transferred. One option is to delete the apps/purchases made by the other ID and download them using the new ID if desired. If necessary, check your purchases history to see which apps/purchases weren’t made by the current ID. 

Purchase History Review.   

Or you can try to set up Family Sharing  between the IDs to keep access to your purchases.

Family Sharing.         

Family Sharing - Set up.