I've recently started jogging, during which I keep getting headaches around my eyes and nasal sinuses, eyes watering. Should I get my eyes tested?

Not eyes' fault. Sounds more like you are having symptoms of hayfever (or seasonal allergies). Consider trying taking Zyrtec (cetirizine) (or generic) once daily, at least 45 minutes prior to when you plan to go out to jog and see if that improves things (if running in the early am, can also take in late pm / bedtime). If doesn't improve, see your Dr for once-over.
Exertional headaches. You may have "Exertional Headaches." Your eyes are likely just fine. However, you may be suffering a form of migraine or migraine like headaches that should be checked out in my opinion by either a neurologist or headache specialist since other things can cause these headaches and need to be ruled out. Check out this link: http://www.headaches.org/education/Headache_Topic_Sheets/Exertional_Headaches.