I know the feeling....been there, done that, and got the T-shirt. Only thing I can say is try and knock out the highest runners/problem childs first. If you can run some tests and make the part run better, faster, and with less defects it might give you some time to work on the lower runners. So long as the part isn't a big pain to get going you should be able to run a process study within a few hours. Getting dimensions on parts for a cooling time test and DOE might take longer, but maybe you can get the QA department to help you with this? Maybe talk to them about how in the end it might help them with less rejections to deal with in the future.

Also if you can work on proving to management that detailing a process and doing the studies helps quality, cycle time, and their bottom line ($$$$$) they may help with scheduling so you aren't so overloaded. I have in the past taken problem child parts and high runners shown how much money they have saved by just taken time to set them up right that it blows their mind. makes it easier to get help after that.