first, here is a page with a bunch of info for your radios:
http://www.cbtricks.com/radios/cobra/89xlr/index.htm
there could be many reasons for a radio to not transmit.
I don't think there is a relay in these radios, but if there is, check that it is switching over when you key the mic.
after that, its usually a bad driver or final, or the radio is out of lock. (assuming that the problem is actually inside the radio and that you aren't just trying to use the same bad mic to key all three radios)
Is there an RX/TX light on the front?
if so, is it changing from green to red when you key the mic?
this will tell a lot about where the problem is.
the truth is that these radios are just too old to be reliable and need to be re-capped.
all electrolytic capacitors in the radio need to be replaced as they are long past their shelf life.
if you don't do that, then no repair is going to make those radios reliable.
even if you do fix one, it is just going to break again as the caps age.
its just the sad truth of our hobby.
these radios were never manufactured to last this long, and they start getting finicky as time goes on.
best of luck with them.
LC