Can pollen cause such bad allergies that it can make you dizzy and give you headaches?

Sinuses. Allergies can lead to sinuses becoming congested which can cause headaches. That "clogged" sensation can feel like being lightheaded.
Dual effects possibl. Pollen by itself can be an irritant like pepper or smoke & cause irritation to the respiratory tract outside of true allergy. Add allergy & you can end up miserable. In my area we have a common affliction in town when the ceder pollen is flying."cedar fever" can do just that, give some flu like symptoms & difficulty functioning.
Sinusitis. Nasal allergies and congestion can cause sinusitis with postnasal drip and headache. Tender sinuses and ear problem causing dizzy spells and imbalance should be treated with 10 days antibiotic.