An increase in HR/BP on DXM is to be expected, it is not something that needs to be treated. It's more annoying/noticeable on those lower, subrecreational doses. Most folks don't enjoy DXM at any less than 300mg at once, that's when you start getting some real dissociative effects and the unpleasant physical side effects are largely relegated to the come up (some folks get a comedown as well). Of course, the higher your dose, the worse the body load is gonna be on the come up, but some of us feel that the good parts are well worth it.
Don't go messing with alpha adrenergic antagonists over something like this, it's just the price you pay for using an OTC drug. You can try to find more recreational, cleaner feeling dissociatives if you are able and inclined to do so.