[T77911S]

car alarms are notorious for causing problems. first you have some kid doing a hack job putting the thing in, then you are cutting the already overly stressed start signal to the starter. a lot of current is required to pull in the solenoid so anything that adds resistance reduces current.
i found the bad relay connection with a test light on my borthers 88. same problem, he thought he had a hot start problem. i tested the start wire at the starter. the light did not shine as bright as it should, or you can use a multimeter. check for 12v.
his turned out to be a lose crimp at the relay that disables the starter. i f it was me, i would not disable the starting or running of the car, just make it so the car alarm sounds. if you have crimped connections, i would also solder them.

you also need to clean all major battery connections, both positive and grounds. dont forget the ground from the body to the engine.

the ignition switch can also cause your problem.

he ended up replacing the starter and the ignition switch for a bad crimp connection.