I've finished quests for Sorine and Gunmar and they both telll me to see Forentius for an assignment. He has the arrow over his head but doesn't give me an assignment. He only shows 3 speaking options, first is "What do you know about the old Dawnguard?", second is "what do you have for sale?", and third is "can you teach me about restoration?". I've waited 24 hours, left and came back after doing something else and nothing has changed. Any ideas?

Check your journal. Do you have a quest active named "Rescue" already?

Another possibility is that you're supposed to be getting the "Lost Relic" quest, but the relic in question might have been assigned to a dungeon you recently cleared and hence, isn't there. Try waiting for 30 days.

I didn't attempt "Rescue" until after doing quests for Sorine and Gunmar and I went to where I was supposed to "kill Master Vampire" but there was no master to be found

So, the Rescue quest is still active in your journal?

yes still in journal

Okay, well that's why Florentius has no assignment for you. Because you still need to complete the existing one. I'm not sure if this is fixable at this point. At least, I wasn't able to fix it.

I'm assuming there was no NPC in Morvunskar to be rescued when you went to find the master vampire? If not, then the only thing I can think of (besides reloading from an earlier save) is to try waiting for 30 days and see if that resets the vampire and the NPC to where they're supposed to be.