Although I cannot find the earlier post using Search, didn't we do this subject already, a few days ago ?
When you say "will only work correctly when download cable is still connected", can you explain exactly how it doesn't work, what exactly doesn't work, when the download cable is connected.
Does everything other than ADC work when the download cable is unplugged ? Have you done any additional testing to check it's not your software partly causing the problem ?
As someone ( dippy ? ) suggested; is it a case that it doesn't actually work with the download cable connected. Inserting the download cable will alter ADC readings unless the Enhanced Download circuit is used.
It sounds to me that you either have the 10K/22K download resistors wired incorrectly or missing, a Reset pull-up missing, or the way you handle ADC readings makes it appear not to work when the download cable is unplugged.
There is no need to connect -9V to the serial input to make a PICAXE work as it should.