Oxidation aside, as brewchez stated.
Yes you can recarbonate the beer. However 30-35psi at 38° just over night probably won't be enough. Just is not enough surface area and time.
Here's some tips
Full carb in a few minutes. Attach the co2 at x2 psi you desired volume calls for IE 2.4 vol = 10 psi @ 38° so use 20 psi. Lay the keg across your legs, turn the keg so the gas inlet is on the bottom side and rock the keg gently end to end, moving the head space bubble up and down the length of the keg. You can hear the co2 bubbles going in when the slow or stop rock it again. Repeat until the bubbles have stoped or 30min has passed which ever comes first. Release the head pressure and attach for serving at serving pressure.
In the Out, no rocking if using a corny ball lock keg, your white gas coupler will actually fit both in and out ports. So attach the co2 to the out port to allow co2 to enter from the bottom of the keg. This gives the co2 much more exposure to the beer and can carb in 1/2 the normal plug and forget method time. There is even a special lid you can buy that has an airstone tube for this which can carb even faster. This is Not for dumbasses. You must make sure this is the only keg on the co2 manifold or disconnect it when pouring from other attached kegs. The pressure drop from the pour will suck beer into the manifold from the keg attached to the out port.
Plug and Forget Just attach to serving pressure and allow a few days undisturbed to carbonate. This is my preferred method as it does well to clairify the beer too. You can cut this time down by laying the keg on its side so there's less beer depth to saturate through. But laying on side doesn't do well for the beer to clairify.