I had some acute achilles pain last year -- it turned out to be related to a twisted ankle, in which I popped a small bone (talus, I think) out of place. Chiropractor popped it back in and pain magically disappeared. I only mention it because I remember that you fell and twisted your knee/ankle a few months ago (not sure when the achilles pain started).
Glad to hear that you're going to see a PT to get ahead of this. Just a FWIW on the down dogs. They are my favorite way to stretch the calves and hamstrings, and I was doing them after every run in the 6 weeks before I injured the Achilles. Not saying they contributed, just that they didn't save me. And, after injuring it, I found out that the trick was going to be effectively stretching the tight calf without pulling on the Achilles, which isn't very elastic. Rolling or massage (the Stick, e.g.) seems safest. One recommendation was the wall stretch, but that feels as if it pulls too much. It sounds as if you're not injured yet and just having warning signs, so any kind of stretch might be OK. But I'd ask if they don't address that.
The heel drops work, not by stretching, but by breaking down what healed up like a tangle of spaghetti, and rebuilding and strengthening. That's why there's a certain amount of discomfort. But it does work if you hang in.
Hope they get you on a healthy track so you can continue to get into the best possible shape for upcoming adventures!