I set up a new LLC in a new state; how do I update my accounting and book keeping?

While there are some gaps in your question, I’ll try to answer.

Assumption: You didn’t change your IRS TIN. You “closed” your books as of the date you terminated the LA LLC (easiest was at month-end).

Under this assumption, you don’t need a new set of books. However:

Issue 1 is the IRS. Since you kept your TIN, all you need to do is change your address they have on file (which can be done on the tax return) and file normally.

Issue 2 for Louisiana is a little more work, but not insurmountable.  Since you notified them that you closed the LLC, you just need to file your last return, finalized Sales and Employment tax accounts.  Everything is based on Jan 1 to the date you told them you closed.  You also need to close any other Nexus States that you may have opened (and pay any taxes due, file final returns, etc.).

Issue 3 is Arizona.  You need to open new sales tax accounts, probably in the same states as the LA LLC, and you’ll file your State income tax from the date you opened your LLC to year’s end.