Frame rates are determined by graphics cards and CPUs. Only changing a GPU will have some effect upon fps but is not the sole determiner; CPU is responsible for the base details while GPU fills it in (generally speaking). As your CPU hasn't changed then the extremes of theoretical high and low frame rates remain unchanged.
This benchmark shows even an i7-5930k with a GTX 1050 ti registering a lowest fps of 80 and average of 102.
http://www.gamersnexus.net/hwreviews/2658-gtx-1050-ti-and-1050-review-benchmarks-vs-460-more/page-5
By comparison, a dip to 40fps for an A6 isn't that bad.
As you've already changed settings with no positive results then it suggests your CPU has reached its limit, ie. being the main bottleneck in your system. With my x4 760k, dropping the resolution should gain you some fps. It doesn't sound like you have so, again, it sounds like a limit has been reached.
You could consider something like trying to lock the fps to 30 to ensure smooth framerates if a consistent 60fps is unachievable. I believe it's something like adaptive v-sync in the nVidia control panel. Or if there are settings which enable this in Overwatch itself.
For completion's sake though I would suggest you check your thermal margins with AMD Overdrive just in case to eliminate overheating issues, update drivers, verify game cache, and close all unneeded programmes before playing (some games are more sensitive than others), and scan for viruses and malware. (Or not if you don't want to.)
As far as I'm aware, the only real way of increasing fps is to have a CPU whose baseline fps for a particular game is higher than the one currently owned. This depends on one's expectations, of course. For the system you have, any CPU upgrade is rather limited plus the FM2 socket is end of life (if not dead). It's difficult to recommend any hardware upgrade you could make to boost performance.