Oh girl, I totally feel you, discovering those bulging veins out of nowhere is such a shock, especially when you were already doing the right things like getting up to walk every hour! With my first pregnancy I had some mild spider veins on my calves that really stressed me out, so I understand that anxious feeling of suddenly noticing something on your body you never expected.
From what I've generally heard around, varicose veins during pregnancy are actually more common than most of us realise, the increased blood volume and hormonal changes put a lot of pressure on your leg veins, and a desk-bound job can make it worse even when you're trying your best to move around. A few mums I know swore by compression stockings throughout the day, saying it really helped with that heavy, throbbing feeling, though one mentioned it takes some getting used to in our Singapore heat. Elevating your legs when resting at home, like propping them up on a pillow, also seems to help quite a bit with the soreness and swelling. The general consensus seems to be that for many women, varicose veins do improve after delivery once the pressure on the veins reduces, though some find they linger and eventually look into treatment options from vascular specialists here in Singapore if needed.
Hang in there, you're already doing the right thing by flagging it at your next check-up, and in the meantime these small steps should give you some relief. Do check with your gynae or a vascular specialist though, since every pregnancy is different and they can advise what's safest for you specifically. You've got this!