Alright. Got a chance to get back to it this weekend. She's back in action!
Seems like there were two simultaneous issues:
1) The ignition switch
2) Bad connection on the "s" terminal
A few weeks back, I put the new ignition switch in, and it started up once or twice. I put the old one back in, and then moved on to the starter. I fixed the connection yesterday, and all the dash lights, buzzer, windows, and fuel pump worked. However, when I went to start it, nothing...I realized the old ignition switch was still installed. When I put the new one back in, boom. It fired up, and the problem seems to be fixed.
Hopefully someone else can see this an save themselves some time if they have the same issue.
Up next: a tune-up, address the slower power steering pump leak, replace transmission pan gasket, and get a new overflow tank for the antifreeze. I'd eventually like to get a set of these helper springs. https://www.hellwigproducts.com/products/search-by-vehicle/?Year1=1989&Make=GMC&Model=V2500+Suburban
And of course, I'm still seriously considering an LM7/LQ4 and 4L80E swap
Thanks for the help, everyone!