I was in the same boat, i has a MSI 270x and i had massive stuttering issues. Finally decided to make the jump to Nvidia for my next card and im very happy with the GTX 960.
The gtx 960 would easly max out the current resolution you have. 1080p monitors are inexpensive but i would recomend a 144hz monitor if you plan to stay in the 1080p relm.
With that said the gtx 960 would go well with the 144hz monitor if you decide to go down that road.