I just want to say, thank you so much Dena and everyone else who gave their insights.
After playing around with what you said, Dena, I found that it was easiest to think of it like singing my speech. I've got a musical background so I know a -little- about singing. This seemed to help take the "edge" off of my voice. I still sound the same, but without that tinny edge. So it helps a bit, at least. I'd give an example, but my throat is quite sore at the moment (fighting something off, I guess).
I may end up talking to the SLP here in my city. They have a free consultation, so I figure it wouldn't hurt to at least talk to them.