Final update: Jeb landed the AAX in the Mun's Northwest Crater, did the science jig, and came home. There was a bit of a hairy moment when I realized the ship was going to land in Altera's highest mountain range instead of the plains. The moment got hairier when the chutes didn't deploy, since the altitude wasn't correctly geared despite the ground being way closer.
In the end, the chutes deployed and the AAX slowed to 10m/s, that last bit being pushed to 4m/s by the last gasp of fuel in the ship's engine before a solid touchdown on the landing legs. Upright landing, on land, with no damage to any parts throughout the mission (except the jettison stages and booster tanks of the AAX which got ditched on a parabolic trajectory into the Mun). Total science gain 1600, and nuclear engines and larger craft are now unlocked, along with better science gear.
Next: to Eve!
And all this after Jeb had been stuck in orbit on the AA III, which was designed only to orbit and survey Kerbin. He got stuck following a glitch where he couldn't re-enter the pod, and was forced to drift along with it for close to a week before I took control of him again, hoping the power on the ship had died. He had drifted 1.5km from the ship at this point and I had to pilot him with his EVA pack (which was at 50%) all that distance so he could get back in his ship, return to Kerbin, and fly the AAX.