If your clock is still working your battery might be OK. I have not heard of any problems with Ultimas but I have seen new batteries die within two weeks. It is unusual but have had it happen to me.
When I think I have a bad battery I usually try the headlights first because they are on their own circuit and give a good load test.

What is the voltage now? If it is low jump it with the loaner car and check the voltage to see if the altenator is working. You should see > 13 volts. If the altenator is OK then since you pulled the fuse to the "Alarm" you can rule it out.

dkveuro has some great suggestion on battery drain. Also look for trunk, hood lights that do not go off or just pull the bulbs until you figure it out.

I do not know if your battery is in the trunk or not but I had a weird problem with a jag xjs that the starter would not engage that turned out to be a bad ground to the frame. I did get dash lights though.

If you cant figure it out maybe you could trade it in on the Camary

Curious about dkveuro comment
AATFWIW....Ignition coils seem to fail on these more than I like.
Is the fail mode such that it drains the battery? I have not heard this one before.

As far as the alarm see if there are any local installers. They may be able to identify your model or tell you how to reprogram it.

Hope this helps. Let us know.