Final Update for anyone who may have this same problem...
Sharp sent out a local Technician today who replaced two internal boards. There are only 3, I believe. The one they did not place (surprisingly enough) was the board with the Ports on it. And replacing the other two boards fixed my set! I can now use the HDMI cable and switch back-and-forth between SD and HD channels without losing the picture sync. Everything works great.
A word on Sharp's call-in Tech Support...
1) They were just great on the phone, and very sympathetic -- until I got to the point of almost demanding a repair. Then Tech Support started to insist that it was NOT this set but the FIOS Control Box, instead (made by Motorola). I guess this repair proves that was not the case (for me, at least).
2) During three different calls to Sharp, their answering system cut me off all three times. Typically, when they put me on Hold to go call the repair tech, or to check the status on something, they could not pick the call back up. I would suddenly be disconnected. To their credit, however, they always followed-up with contacting me back by a return call, or the Tech would call from the Repair Center.
The Technician that arrived said that Sharp usually insists that this set be brought in for service. The one advantage to doing the repair in my home was that we could replicate the condition before the repair -- then test it immediately afterwards. He had never seen such a problem before, and swore that Sharp knows nothing about it even though there are quite a number of incidents reported here and elsewhere. There were a ton of screws he had to remove to get the back off, including the TV Stand.
But, thank you Sharp! The set works great at this time. Oddly enough, when I asked the Tech which sets he liked he said...
a) Panasonic -- because their Service Dept. stands behind their products so strongly. Both he and his assistant repeated this many times.
b) JVC -- next to Panasonic, their hands-down favorites for ease of repair and dealing with the company.
c) Sharp -- because "the sets are great when they work".