Thanks for the additional information.
The straightened paperclip priming method is very old, and we now use a small syringe with plastic priming tip attached to prime the exit channel with ink. The syringe method is cleaner and easier to do, and is better for the cartridge (some people have damaged their carts by piercing the outer rubber exit valve seal with the paperclip).
If ink doesn't flow out of some carts, it may be due to infrequent use of the printer, and agitation of the carts, causing pigment to settle to the bottom of the carts, collecting in the exit channel and restricting ink flow over time. This would also cause thicker pigment from the bottom of settled carts to be drawn into the printer's ink lines, which can lead to particle build up in the dampers faster than normal (though, dampers should normally be replaced every 1-3 years, depending on use). Blocked damper screens will restrict ink flow until dampers are replaced. If your air vent filters have ink in/on them, this could also restrict ink flow from the cartridge, but based on your use history, I suspect you're dealing with clogged dampers and possibly clogged exit channels in carts from thick pigment due to not agitating carts and using the printer as we recommend. In this case, I suggest replacing your dampers, and cleaning the print head while it's exposed (when dampers are disconnected). Do you still have ink bottles? If so, it may be good to shake your carts and pour ink back into the corresponding ink bottle, then try to flush out the carts using distilled water to open the exit channels, and if successful, get the carts as dry as possible, then refill with ink (preferably fresh/new ink).
You may also need to replace your dampers, which are easy to do with the 4800/4880 printer model. You can get the repair manual online, one place I know is www2manuals.com, which gives clear step by step instructions with detailed photos that are very easy to follow. We have replacement dampers for your printer model here: http://shopping.netsuite.com/s.nl/c.362672/it.A/id.5442/.f
I hope this helps, please keep me posted.