Are your fog lights on? Will they come back on if you just turn you fog lights off? If so, it is probably just the foglight overheating problem. You can run power directly to the fog lights through a relay from the battery and use the existing wires to control a relay.
Another possibility is your mult-function switch. When the lights go out, try jiggling the turn signal switch. If they come back on, replace the switch.
Yet another possibility is a bad headlight switch. Ford did a really bad job with this stuff since it all tends to overheat. If you pull your switches out related to the headlights you will notice that they are melted.
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351W 89 Mustang GT Convertible