I went through my saved passwords and deleted the ones for that account, but it still tries to log me in to that google account first, even though firefox doesn't remember my password. Because it's google through an external site, there is no account-swap button available until I clear my organization's secure login page. Is there a google account priority somewhere that can be changed? What else can I try to clear?

This is my personal computer. For a while, typing a google webpage into the address bar used to take me to my personal account first while I was using both accounts, and at some point it switched to taking me to the organization's account first.

Ah ok, that was a good suggestion. I think I figured out what is happening. You can be signed in to multiple google accounts at once and they stay signed in after restarts and shutdowns, and whichever one you sign in to first is your default. The only way to change default is by signing out of everything, then signing in to the one you want to be default first, like you suggested. At one point, I must have done a sign out/in and chose my company account first, so it has stayed my default for the last few weeks. Now I'm signed in to both accounts and my personal is my default, and it stays that way after closing firefox or restarting my computer.

Thanks for walking me through!

I think the method of deleting your work account entirely. Once that is done, login to all the different apps and website pages you use with Google login. Make sure he's using the personal account on each one. Once that is done, you should be able to re add your work account to those pages and have it remain as the secondary account. Also worth checking is in your phones settings page under accounts, make sure your personal account is your Google account. These steps should solve the problem. It would be nice if Google would add some type of default account check that would make sure these stayed the same. I have never seen that option however.