OP, the original bulbs were probably incandescent or halogen both of which tend to run hot. In many cases you can find LED replacements bulbs with run much cooler and much lower power requirements. I would look at the original bulb for a model number, then look for a LED replacement. I would then carefully clean the existing bulb socket, and install the new LED bulb, with any luck you will be back in operation. Since it appears you have to original bulb socket out of the unit, I would be glad to take a look at it and possibly clean it.
Send a PM is you are interested and would like to discuss over the phone, NOTE I screen all calls, so you may need to leave a message for a call back.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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