Yes, I have successfully made this transition. Here is how and why:
When I started running I was a heel striker and I suffed from shin soreness all the time. I assumed the solution was to get a decent pair of trainers so I went and got my gait analysed, was established as an over-pronator, and sold suitable shoes. While they did help my shins, they did not rid me of the problem entirely.
About three and a half months ago, after much research, consideration and evaluation of my running technique I determined that the shin soreness was most likely caused by heel striking and overstriding. I made a deliberate effort to shorten my stride and to also adopt a mid/forefoot strike. For a few weeks I mainly did some very very slow runs in order that I could concentrate properly applying these - and other - techniques. The shin soreness pretty much disappeared altogether, but now, like you, I found that my calves were always stiff and sore after a run.I made an effort to stretch properly after a run, and also to stretch once or twice on non-running days. I also (grudgingly) rested a little longer between runs. My calves still get a little stiff, but do not really get excessively sore anymore (unless it's been a particularly hard session, but then it is to be expected)
What is interesting is that I went and had my gait analysed again a couple of weeks ago, and I now have a neutral gait, and so I bought some neutral shoes. Since running in these, the ankle soreness I had has also disappeared. I assume this was caused by running in shoes that were no longer suitable for my gait.
So, to summarise: a) you will get used to it in time b) stretch, rest and don't overdo it c) it's worth getting running shoes suitable to your new running style