Made the mistake of trying refill carts from my local Walgreens and immediately realized they weren't going to work for me and returned them. I installed new genuine HP carts, but ever since I have experienced a significantly slower response time between sending print command and actual start of printing by the 7960. I've tried a complete reinstallation, but still have the problem. Any ideas on how to correct will be much appreciated.

Do you have the old, empties around...preferably enough so you have three of each?

That series of printers stores the cartridge information for three cartridges...so in order to wipe that memory, the easiest way is to insert three cartridges, power cycling between each to make sure it registers. Right now, the memory is probably holding the empty, the cheapie and the new one. An older empty would push the last empty out, leaving room for the latest empty to push the 'refill' out. Then when you put the new ones back it, it should register as full and then stop running the 'check levels' routine every time you print...which probably where your delay is coming from. The HP software/drivers will check the levels every time you print when the ink level gets to a certain point (not quite empty). With new cartridges, it doesn't check as often. I've found this out pretty much by trial and error with my HP printer (a 5650, which is a very close relative of your 7960...less bells and whistles...same basic printer).