Update:
Came home with my buddy's hornady dies. Seated a few bullets...still shavings. Following my friend's suggestion I then tried something else...He suggest I take some fine steel wool and smooth the inside of the case neck where I had chamfered. I didn't have steel wool so I took the little sharp edge of of my caliper and ran it along the inside of the case mouth to debur it. Bullets seated noticeably smoother and no shavings! I pulled the bullets and with light shining on them you could see barcode-looking marks made by the case mouth but running my nail along the top of it I couldn't feel an edge like I could on the pulled shaved bullets.
Also interesting is I took some cases that I prepped a while ago the way I always do...trimmed and chamfered (without deburring just on the outside of the case mouth, not the inside)...and no shavings.
I'm wondering if deburring the inside of the case neck is just a band-aid and isn't the real problem though. I say this because:
a) I loaded 500+ rounds chamfering the same exact way...chamfering with only deburring the outside after trimming...not deburring the inside of the case mouth....without a problem
b) my friend prepped 10 cases, chamfering all of them. He loaded 5 in his press with my dies and they were perfect. I took the other 5 home and loaded them, and shavings.
c) I took new brass, resided just the neck, and did not chamfer them. Got less shavings...but some still there. If the problem was just with my chamfering tool or the way I was chamfering then new un-chamferred brass would have no shavings.
This is the chamfering tool I use. Maybe I should get a different one and try?