I would recommend a shoe with a higher drop to reduce the stress on the Achilles. Adding a removable insole and cutting off the forefoot (in other words creating a heel insert) can increase the drop further to take even more stress off your Achilles. I did something similar when I was having trouble with plantars fasciitis and it worked wonders. Also, I would recommend a stable shoe that is relatively stiff to minimize the amount of Achilles activation while running. Shoes like the HOKA ONE ONE Bondi or Gaviota could be good options to enable you to run comfortably and avoid further Achilles irritation. I hope this helps!
Important Caveat: I am not a medical profession, nor particularly knowledgeable. My two-cents here is based off of an internet article and one successful training cycle without pain.
I have experienced mild-moderate achilles issues in the past, oddly with the Mizuno Wave Rider 20 (a high drop shoe). I came across an article that recommended rocker geometry shoes as a way of dealing with this issue and last fall I used the Saucony Endorphin Shift. The Shift has a mild drop but a crazy rocker geometry and I had really great success putting a lot of miles on them without any injuries or pain. Of note, the Shift is also fairly firm as compared to many shoes but I can't argue with the results (even if they are possibly unique to me). The first generation Shifts are on sale for about $100.