Methyldopa is actually one of the more established options for blood pressure in pregnancy, so your doctor is following a pretty standard approach, that said, the trade-off most mums I know mention is the drowsiness and dry mouth, especially in the first week or so while your body adjusts. Someone in my mum group mentioned she felt quite lethargic around the same stage and had some mild stomach discomfort, but it settled after a few days and her BP stayed well-controlled until delivery. On the more serious side, a few mums flagged to watch out for things like unusual stomach pain or dark stools, which would warrant calling your doctor straight away rather than waiting. Overall most found the side effects manageable versus the risk of leaving blood pressure unchecked at 36 weeks, but since everyone's body responds differently, it's really worth having a proper chat with your own gynae or specialist about what to expect and when to flag concerns.