Printer Ink Cartridges & Print Quality

I've been having a recurring issue with my black ink cartridges (the standard 63 type) not working after only a few uses over an extended period of time.  I typically only print something out once a month or so and initially thought maybe the ink was drying up or something after sitting unused for too long but I don't know if that's even a thing.  The printer display is showing the ink levels are still pretty much at full and there's no sign that it's leaking out anywhere.  I've gone through 3-4 black cartridges like this over the last year.  Get only a few uses out of them and then they seem to need replacing, which has worked as far as getting the device to print in black again but I'd like to figure out why they're not lasting a lot longer.  Color cartridge seems to be fine, it's just the black ones that keep failing on me.  Also, just to clarify, my product info sticker on this printer says it's a HP Deskjet 3630 All-In-One Printer series but the label on the front of the swing-down door to access the cartridges says it's a HP Deskjet 3636 if that makes any difference?  I also seem to be having difficulty connecting directly to support because it's asking for my serial #, which I believe is the one listed under the barcode, but also needs a product # which I can't find anywhere on this device unless that's the same as the model # but the website doesn't seem to recognize it?  Thank you.