Since you cut them with the sap up I would dry them and then peel them. They might split if you peel them now although the bark will come off easier when green. Put them up in the attic for about 3 weeks and see how dry they get. You could probably peel them then without worry. Also may help to tie them up in a bundle to minimize warping while drying. It gets warm enough in my shed during summer to dry things but if you don’t have a shed the attic works or even a car not being used that gets sunlight.
I use heat to straighten but try not to burn with an open flame. A small rasp and a cabinet scraper will help smooth out the occasional knot and get the diameter more uniform. It’s alot of work but dogwood makes a heavy indestructible shaft.

Duncan’s advice is great, but I’ll add a few things. I’ve never worked with dogwood but have used many others such as sparkleberry, arrow wood, yaupon, etc. All are more or less the same in what it takes to make a good shaft.

I usually cut mine pretty big because they tend to shrink a little when they dry. Also, the bark makes up a good bit of the diameter so you loose a lot when you scrape that off.

I’d do a little pre straightning before bundling and drying. It’ll make it easier later.

ch