I had a mis-diagnosis of an Achilles injury when I messed up my ankle. The PT for that was 3 sets of 30 seconds calf stretches for each: gastrocs and soleus (straight leg and leaning/on tirangle block or stair, slight knee bend and leaning/block/stair).
Sometimes, my Achilles will get tight when I have too much tightness in the hams or calves. But that's definitely a tension feeling and not soreness post-ride. However, if you are too tight in those muscles, it can cause more strain on the achilles while you're on the bike.
Lowering the saddle may not help, because if you bottom out on the pedal stroke too early, you will get hit with some upward force that might not be comfortable for the tendon (the same force that would push you off the saddle if it was far too low). You might just want to think about riding with a neutral akle, neither over-shortening the tendon with too much toe-down movement nor straining through by pushing your heels down far. Generally, though, heels-down (stretch) would be better than heels-up (contraction).
The best way to get it to heal is to alleive tension through calf stretches (as well as the ice/ibuprofen). Massage of the arch and sides of the heel may also help. (The achilles attaches at the rear of the heel and also laterally, near level with the bursa.)