well locking the saddle to the leadscrew is a must, tho a proper saddle lock is better (nice little weekend project to make and fit - works similar to the tailstock camlock... unless of course you've not made one of those yet). Most of your problem there is probably that, tho movement of the cross-slide on its gib might also be a factor if the gibs aren't properly adjusted.

You need somewhere between 600 and 800rpm for that diameter/material, its actually 560 at the outer edge but increasing rpm as you go in, so 1120 at the 10mm mark, so take an average as you can't easily change spindle speed on the fly.

With stainless you need to be taking a very light cut and use lots of coolant and a newly sharpened tool. I can't easily tell from the pic as the tool tip is out of focus but it doesnt look too healthy.