Can spring allergies cause "allergic asthma"?

I’ve got a lot of allergies. I’m allergic to the cat that lives with me, but I take medication every day that makes these allergies subside.

Lately with the change of season and the introduction of pollen (and LOTS of it) I’ve been feeling worse.
My chest is really tight, and it feels like I have labored breathing.

Could this be from “allergic asthma”? I don’t have any running nose or itching eyes, no sneezing, its just my throat and chest feel tight.

I’m thinking about going to my local Doctor on Monday to see about an inhaler. Is this a good idea??