I have no experience with 3rd-party bulk inks on the Designjet 5000 machines, but at the end of the proverbial day, inkjet printing always comes down to the specific chemistry between the inkset and media where the ink is going. If you are using a media that is known to work with the HP UV or Dye inkset, such as an HP photobase paper or Sihl / Intelicoat paper product for aqueous inks, and the ink is not drying, or bleeding, that points back to the ink in the printer.
In that case, I would purge the 3rd-party ink and go back to OEM ink, although I am not certain that OEM inks are still available for those machines, given their age, but I would assume that they are.