New printhead, yes that does sound right since the comment is coming from a company who will profit from that advice. Seriously, though, it may well be the best option. You can remove the printhead and try cleaning it at home in hot soapy water, rinse, then leaving it to dry thoroughly. Certainly won't leave you any worse off than you are now, and may well do the trick.
I future, you must use an inkjet printer every few days to avoid printhead blockages (the driver's nozzle-cleaning routine can only clear moderate blockages, and wastes expensive ink in the process).
Anyone who cannot guarantee that they'll be using a printer every week should not really be buying one as it may prove to be more trouble than it's worth. In such cases, it's often better to ask a friend to print what you need, and offer to contribute towards ink costs.