I fill my kegs with CO2 before filling them with beer. Vent the pressure then fill via the liquid post. Since I ferment in sanke kegs, I use a CO2 push to move the beer to serving kegs. This severely reduces the chances of oxidization. When I bottle from keg/tap, I also make sure to fill the bottles to overflow (clear beer not just foam) and then close/cap them. If you have a beer gun, be sure to purge the bottle with CO2 before you fill, then cap/close quickly. Not being able to do that previously, is why I've been using swing-top bottles when bottling from tap/keg.
Also, depending on how much head space I have, I'll purge a keg 5-10 times once filled. This is with the lid on, using the pressure relief ring/valve in the lid. Not had any issues with this method. I also, typically, use 5-10 psi after hitting the keg with enough pressure to seat the main/lid o-ring.