Big Kureyon does soften up a bit with washing. I've made a scarf with it and really like it

Noro Iro is fairly bulky and stripes also...I'm sure there are a few other Noro yarns that do. Some of them, like Iro, have silk or other fibers thrown in to make them a bit softer.
Lana Grossa Inserto (98% wool, 2% polyamide) and Filatura Di Crosa 127 Print (100% wool) both stripe similarly, in that they have a main background color and stripes of contrasting or matching colors interspersed throughout. Inserto knits at about 3 st/in and 127 Print at 4 st/in...not sure how bulky you're looking for

ETA: Oh! And it's pricey, but Lorna's Laces makes worsted and bulky weight self-patterning yarn (I think it's superwash wool). Soooo soft!