The Wiring Diagrams manual for the car details all the grounds in section 10, then you can find the locations of each using section 151.
Not suggesting it would be easy, but it is possible.
My (non-FGT) experience with electrical problems like this (where there is erratic behavior of multiple components) is that the problem is always related to a loose or corroded ground connection somewhere in the car.
Assuming the new GT radio has a conventional antenna connector, a plug-in unit will work much better than a transmitter-type unit. Still sends via an FM radio signal, but piped directly into the radio instead of through the air.
Here's one example...
My car has the standard (non-Mac) radio, and while I have no desire for a Mac system or any high-power audio system, I decided it might be occasionally nice to have a radio with both plug-in and bluetooth aux in. Virtually all radios are black these days and have flashy multi-color animated...
I bought this $160 neon sign on ebay, looks pretty nice.
https://www.ebay.com/itm/163867715772
(GT is white, the ring around it is blue.)
They have some other Ford signs, too.
https://www.ebay.com/sch/m.html?_odkw=&_ssn=stayzoko-inc&_nkw=ford
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More numbers.
So that Bring a Trailer buyer had a budget of $264,000 for that car, and the seller got $259,000 of that. Pretty efficient. (BaT buyers fee is 5%, up to a maximum of $5k. In this high-price sale, that $5k equalled 2.1%)
In a typical normal BJ/Mecum/etc auction (let’s say 10%...