Car Alarm works accept for both door entry alarms

Check your owners manual or with the installer to see if there is a bypass. It's possible that you accidentally turned them off. Otherwise make sure that there are no shorted wires giving a false closed report to the alarm system.

I think the alarm works off of the dome light switches which are working.

Many car alarm configurations sense door opening just by the battery voltage drop your interior light poses. (This assumes their switch sets them to light when door opens and bulbs are OK)

You need to pull the alarm brain down and and look for the door trigger wires. Most alarm companies use a green wire for a (-) trigger and a purple wire tor (+) trigger. Then meter that wire to see if it is picking up the door trigger. The polarity will switch when the door open vs closed. If you are not getting any voltage then trace the wire to where it connects with your factory wiring. Most installers use a T-Tap connection which can wiggle loose with time or an improperly sized t-tap. However, if the connection is good, the voltage is present at the alarm brain, and the dome light/underdash light comes on its time to change the 24 year old alarm. I suggest a Viper 350. The Viper is reliable when proffessionally installed, sold everywhere so you can keep your warranty, and is moderately priced around 200 USD installed.