I already received the replacement AirPod just one day after I requested it. Now I'm worried the problem might actually be with my charging case though. Yesterday, I removed the faulty right AirPod that won't charge and charged the case. I haven't put the defective AirPod back in yet and the working, left AirPod was fully charged yesterday.
When I went to put the replacement AirPod I received today in the charging case I noticed that the charging case was only at 14%. There is no way it should have been that low considering I just charged it yesterday and only the fully charged left AirPod has been put in the case since.
The replacement AirPod was at 74% when I received it. I put it in the charging case and now both AirPods show at 100% so at least I know this one charges when the old one wouldn't. The charging case is currently at 9%. We'll see what happens after a full charge but I'm really hoping I won't have to have the charging case replaced too.