I've been to Jamaica and Dominican Republic.
Jamaica is beautiful and, in my experience, has more to offer if you're going off the resort. But for a lone traveller, especially the first time, I can't recommend it as the safer or more relaxing option, sadly.
DR felt a lot more touristy and the resort was much larger than the Jamaican one I stayed at. When I think typical beach vacation, this nails it. I would feel safe travelling there alone.
As for time of year I can't say. Both my visits were around July.