The crucial question is: What gas is in the headspace? In the primary fermenter, air/oxygen exposure can't be that bad. Consider the generations of brewers who used, or even continue to use, open fermenters. For the secondary fermenter, also known as a bright tank, air exposure is quite likely to be deleterious. Flush the secondary with CO2 first, and that's not an issue. The OP is using a 10 gallon Cornie, which makes that relatively easy. Flushing a carboy with CO2 isn't necessarily as easy, but certainly can be done.
When I sanitize my Cornies I fill one with StarSan, then push that into the next one with CO2 using a jumper. This leaves the Cornie full of CO2 and ready for use.
Want to use a carboy instead? To fill a carboy with CO2, think of a water cooler. An inverted carboy filled with sanitizer can be set in (large) bucket with a little sanitizer in the bottom, enough to cover the mouth of the carboy. Then just run a smaller diameter blowoff tube from your primary into the carboy, starting as soon as possible after you no longer have krausen coming out of the blowoff tube. The Co2 from fermentation will gradually displace the sanitizer, filling the carboy. The pressure head is only the cm of water above the mouth of the carboy. Doing this you never much exceed atmospheric pressure, and are therefore quite unlikely to blow up your carboy. This approach can be used with no CO2 tank, just a somewhat longer than usual blowoff tube. I costs only a stopper and connector (or Carboy Cap), the length of hose, and a suitable bucket, like those $6 big party bucket. I assume you have sanitizer. Of course, if you have a CO2 tank, you can use the same inversion method to flush a carboy with a hose from your tank.