Yes, I definitely do prefer a bag that goes over my shoulder.
I find these far more convenient as if you sling it right across your body, you have always two hands to do what you want with, especially when shopping.
I find bags with short straps are a nuisance as you only have one hand to "operate" with, so I have very few hand held bags and those I have are evening bags which I no longer use.
Handles are such a pain . . . you always end up with them up in the crook of your elbow and it hurts and you knock things over etc. . . like a guided missile. Ah, happy days, shopping with mother and her handbag. Haha.
All the bags that I have are shoulder bags. I like to have my hands free, as I am quite 'touchy feely' when out shopping, and like to pick up the products and look at them. It would frustrate me if I couldn't do this because of a bag. All my evening bags have an optional strap inside as well, for convenience. It's nice to have the bag on your shoulder where you can leave it, forget it, and continue to eat or drink, or talk with your hands like I do. I think that bags have definitely evolved into more of a shoulder bag these days. It would frustrate me to have to carry something around all the time.
I use normal handbags as I've found the straps on shoulder bags slide off my shoulders which is quite frustrating. I found also that because I'm big busted the cross body bags come under my armpits so my solution when I need my hands to be free is a small security backpack.