After working out some bugs, we've got the Tweets going again from our server - WITHOUT the use of "TwitterMail".
TM was becoming such a pain in the a**. The guy just DOESN'T seem too concerned at all about what happens with his system, and he's not up on keeping people updated with the latest, so everyone would always find Tweets going through TM would just STOP...for DAYS at a time...with NO explanation. Then they'd just start right back up again. And all last night, we discovered that apparently TM was queing bulletins being sent and holding them for later. Only when later came, they BOMBED us with dupes of one particular Zone Forecast Bulletin that lasted from 7:52pm last night until about 5am this morning. That was the last straw.
So with the new means of tweet distribution having been worked out, I have REMOVED TwitterMail from the equation entirely, and we are now sending tweets to Twitter completely of our own. We no longer have the advantage of TM's HTML buffering of our bulletins, which made long bulletins prettier to view, but we can still get them out in multiple 140-character "packets". I would rather deal with this than with TM's LONG-long downtimes and tweet bombings, and reading the owner's stupid comments on his contact page.
The Local Storm Reports were a nice convenience which viewed well with TwitterMail; but unfortunately they don't format well under Twitter's API interface, so I'm going to stop sending those. If anyone REALLY wants them, I can send them to your email.
At any rate, looks like we've solved the problem with TwitterMail.
Barring Twitter itself going down, we look forward to much less downtimes or problems, now.