I think your 944 cartridge is a Lexmark 31/33/35 in disguise. I see the Dell cartridges cost a huge amount in the UK, if that's where you are. You may want to check if a Lexmark 31/33/35 looks as though it will fit, and if it's cheaper. Prices are way, way better in the US and Japan.
Anyway, the Dell should be an easy refill. Neither it nor the Lexmark are chipped. I never had much success refilling HP cartridges: if the ink starts coming out the print head, it wicks up into adjacent 'oles.