i don't think you're going to have a problem. as long as your beer stays cold, reactions are pretty slow. as has been said here, skunkyness is a product of uv degradation. the worst you might get is some minor oxidation, but even that, if you leave the keg closed, there will be a nice pillow of co2 on top of your beer which should protect it for a while until you can get a new tank. recarbonating your beer is not a process that will produce off flavors.
fwiw, whenever i put a new keg in the fridge, i pressurize it, usually to about 30psi as i find this helps to set the seals. then spray down the top with starsan. i look for bubbles that might indicate a co2 leak, as well as listening for faint hissing sounds coming from the keg. once that is done, you can vent the pressure and bring it back down to your carbonation temp.
lastly, keg lube works wonders. i've got a couple kegs that refuse to seal properly without keg lube. and i mean, they don't seal at all. turn on the gas and its so bad your hair starts blowing around if you're anywhere near the top of the keg. 30psi and keg lube will fix that right quick.