If I buy a larger boat I assume it will also be higher/taller because of a larger cabin, possible flying bridge or mast for the RADAR.
Easy… First you need a VHF-Marine Radio, almost always found on used boats and less than a few hundred dollars if you need to buy a newer model. Call the bridge on either channel 13 or 9 (some states vary) to request a bridge opening.
But before calling the bridge you must know how tall your boat is and check the charts for bridge clearance and the tide level to see if low tide gives you more air space beneath the span.
An additional asset is the number board at the entrance of the drawbridge showing actual clearance at the moment you want to go through.
Some bridges have schedules that only allow them to open on the top and bottom of the hour (or quarter to and quarter past) and even then will only open on request.