I'm almost afraid to say it for fear of jinxing everything, but I think I have finally fixed my problem. Thanks to Sir James for reminding me to tinker with the NIC drivers.
I've been running STO under Windows 7, using the drivers that are installed with Win 7. I downloaded an older version of the NIC driver (don't recall the version number, but it was a 2008 release). I was able to play for three straight hours last night, during my biggest problem time, without so much as a "SERVER NOT RESPONDING" warning during zone changes.
It appears that STO was having trouble with my Windows 7 network driver. I really hope this helps someone else who might be having the same problem.
Something else that occurs to me for other who might be running into his problem. The driver that I rolled back to (from the default Win7 driver) was of course downloaded directly from the manufacturer's website. It is my understanding that most of the drivers that shipped with Win7 were submitted by the respective hardware manufacturers, but in my case the network driver installed by Windows was generic. If "un-updating" your network card drivers doesn't fix your connection problem, consider looking for other generic drivers that may not be optimized for your hardware.