Seeking advice: using handspun yarn in a hall runner/rug

Hello folks, I've taken up spinning as a "why not" activity and have some questions about weaving with the resulting yarn. I don't trust myself to spin finer yarns yet, so my plan is to plain weave a 2-ply suffolk/suffolk cross weft of 8-10 WPI on the cotton warp I've already got set for rag rugs. My specific question here is whether I would be better off waiting on fulling the yarn until after weaving is done. This is apparently how it's usually done with handspun, but that feels peculiar to me? Also, if you've got any advice for me that I may not have thought of re: weaving with my own handspun, please let me know!