Yes your gut feeling is right.. you don't want to create hypoxia in your culture medium.

Here's a makeshift design you can easily do yourselves with some lab disposables lying around if you want more oxygenated cell culture medium. You can create a very simple DIY oxygenator (uncontrolled) to "aireate" your medium (not controlled but very effective). Use silicone tubing (the good stuff non cytotoxic etc), block one open end by sticking in a rod (~ 1cm long) (metal or plastic will do as long if it is autoclavable, make sure it's sealed tight), perforate the tubing just above the cap with a small needle (a few holes will do). Thread the tubing through the bottle cap. You either make a hole yourselves, as long as it is slightly smaller then the tubing, you want a tight seal preventing infections, or better get a proper connecter cap for it. You stick the capped end into the culture medium, connect the other end, coming out of the spinner flask, to a pressurized air valve, don't forget to connect a 0.2 micrometer filter in between to make sure the air is sterile. Bubble air through your medium slowly, you don't want to create foam or shear stress, by too much air buubles going into the medium. You'll be "sparging", that's what it's called, with ambient air, but at least you have plenty of oxygen in you medium. Last but not least you need to make sure that your caps can release some air from the bottle, loosen them a bit, you don't want to build up pressure in your spinner flask. If you want to be fancy you can stick in an oxygen electrode to continuously measure oxygen, or hook it up to controlled gas mixture but that is a more expensive option.

Have fun..