Can your carotid artery spasm?

Not really. Carotid disease cause stroke, only fmd (fibromuscular dysplasia) causes muscular spasms but they are typically asymptomatic...
Yes. . Any artery that is not "hardened" with plaque or calcium can go into spasm. However, connecting spasm with clinical symptoms is a more difficult. If you are concerned you should consult a vascular surgeon or neurologist.