Does an increase in blood pressure and pulse (due to prescription side effect) cause flushing/feeling intense heat? if so, why?

Maybe, but the main: issue is dilatation of arterioles within the skin resulting in increased blood flow in skin capillaries; in turn perceived as flushing, a feeling similar to heat exposure because both (a) the capillaries carry more internal warmth to the skin & (b) the sensation matches ↑skin blood flow (to protect/keep skin temperature lower) when we are exposed to heat from outside our body. ↑BP may accentuate.