Hmm. I'd replace the ignition switch first; the fact that those two things happened together is probably *not* a coincidence, and the switches have a habit of failing (without warning; many here carry a spare with them so that they can swap it in when needed.)
As it turns out, the A & B solenoids are interchangeable. While it might be messy to drain the fluid again (if you don't have a drain plug), it would cost you nothing to swap them, and see if the problem follows. It's possible that *both* solenoids could have gone south, but that's pretty rare. As you know by now, the A / B solenoids aren't terribly expensive; you could just throw a new one in the other position and save yourself a pan drain (or, swap them, and if that *doesn't* resolve / change the issue, take the new one back w/o installing it))
What's likely, since you're starting out in 3rd currently, is that you're in failsafe / limp-home mode (b/c 3rd gear is the only forward gear that doesn't require electrical power). Check the wiring harness for an open / short; if you find one, replace the harness (the wires terminate into a round plug on the outside of the case on the passenger side, and it can be difficult to get that plug out / in, with the trans in the default location (e.g.; you might have to loosen the rear mount, bend the shield on the passenger side of the case, or drop the trans outright to get to the plug))
Since you can (apparently) shift manually, the problem is most likely electrical in nature (which again, implicates the ignition switch, if you think about it). Use 2 or 1 to start out with until you find the cause & get it fixed. (doesn't matter; you'll be in 2nd by default, and you don't have 1st right now)
Here's a pretty good comprehensive guide to common 4L60e issues & fixes. This page states at the top re: fuses & to check them first. That's the cheapest / easiest solution of all, of course. May / may not apply (this is on a an F-body section of LS1Tech, although it was lifted from a GMT400 site, which is the 88-98 full size pickup (C/K series).