Thanks again for all the welcomes.
I thought I would post a little update. As I mentioned, I submitted my 107 application to IACRA and it was displaying a status of "Signed by applicant." It has been "Signed by applicant for a bit over 3 weeks now. No changes in that time, no emails from TSA, nothing.
Today I finally gave up waiting and started calling the FAA to see what was going on.
After 6 phone calls, 7 FAA employees, calls to the FAA Help Desk where they could see the application and it looked fine to them and sent me to the FSDO. Calls to the FSDO who couldn't even see the application and thought I needed to come in person to do a 107, then sent me to the Airman Registry. Calls to the Airman Registry who again couldn't see an application and sent me back to the help desk. I was eventually able to talk to a gentleman in 2nd level support who said, to paraphrase, "Yeah, your application looks fine but it is stuck in IACRA for some reason and I don't think it will ever process. We could try to figure it out, but you should just submit a new one."
So, I submitted a new application, which immediately went to "completed" status. IACRA now has a link where I can print my temporary airman certificate. Certificate Number: PENDING. I thought there was going to be a time delay on being able to print a cert but perhaps the TSA check I went through 14ish years ago when I took some flying lessons still applies? Or perhaps the TSA check is was started on the 7th? I don't know, maybe everyone can print a temporary cert immediately after submitting the application.
Edit: The status changed to "Transferred to Airman Registry" the next day so I think the TSA checks must have already been completed in my scenario.
My advise for others: If you are submitting an application for a Part 107 certificate and IACRA gives you a status of "Signed by Applicant", something has gone wrong and you should try again with a new application.