I know someone (Cosh?) was having problems last night, as was Porky, but I'm pretty sure neither of them have the TS3 overlay plugin installed. I don't know anything about the inner workings of that particular plugin. I'm assuming that
this is the plugin you're running. If it is, it probably uses some sort of root window hijacking trick and is possibly not something SWTOR likes. On the other hand, it could be completely coincidental. The only way to know is to keep it removed and disabled. Although, I'm not sure what you mean by "accessing TS again jacked something up;" does TS now seem to be the source of your problems? If so, it may be due to that plugin which might only be resolved by a clean install of TS.
If you reinstall TeamSpeak 3, make sure to go to Settings -> Identities -> and click the "export" button first in order to retain privileges on the TS server. If you don't (and specifically, if you don't reimport that), TS generates a new identity which it sends to the server, and it thinks you're a new person.
Judging by what you posted, I can surmise that the first helpful link you came across was
this one which suggests to uninstall anything you're not using (he mentions WinPCAP, which isn't going to run unless you're running something like Wireshark as the libraries are never loaded until needed), but that seems a little... excessive.
This post does bring to mind something to consider. You might want to verify that the SWTOR binary is in your firewall rules if you're using the standard Windows firewall. It's pretty easy to check. If the problem returns with the overlay plugin removed, I'd check there first. AFAIK, Windows takes a checksum of the binary, and if that changes (such as from a patch), it may not have re-added that. I can't think of an instance where that's occurred, but I'd imagine it's possible.
Though, I can't help but think based on what I overheard in TS last night that it's a server-related problem, because two of our guys were experiencing it last night and couldn't reconnect for quite some time. Moreover, the error ID would periodically change.
You may wish to seek out Cosh and ask him about it.