Try something like this: http://www.fatbobspaintball.co.uk/paintball-clothing/paintball-jerseys/eclipse-hde-jersey-camo.html#.Vp_CEvmLSt8
It's padded in the chest, shoulders and arms. It won't STOP you getting welts but it may well help to mitigate them. There's trousers etc too.
I get bleeders more than a lot of my mates, don't know why but there it is. I have a customer facing job so I try and make sure that if I know I have something coming up then I wear something more likely to give me protection. My UBACS (a Viper one) has fairly heavy cotton sleeves which help but I have taken the foam pads out as they make the arms stiff and I hate that. I run a WAS 901 chest rig with the back panel so I don't get too many hits there either but there's always going to be spots where you get hit that will welt up (side of the chest, inside the leg/arm). If it really is a serious health issue for you then maybe change your play style - use cover more effectively and simply try harder to not get shot.