Hi Sabo,
I was in the same situation as you last year (not twins, though -- a singleton. William was born healthy & happy four months ago). You can actually ride a lot of things, but nothing that involves great speed or drops, or motion sims. Avoid:
MK - Space Mtn, Spash Mtn, Big Thunder Mtn, Mad Hatter's Tea Party, Tomorrowland Speedway
Epcot - Test Track, Body Wars
AK - Dinosaur, Kali River Rapids
MGM - Rock 'n' Roller Coaster, Tower of Terror, Star Tours
All of these have warning signs in the queue areas.
Other tips: stay well hydrated, carry healthy snacks with you (I carried a little package of dried fruit with me at all times, because I'd be feeling fine and then without warning I HAD TO EAT, IMMEDIATELY!) and rest frequently. Don't overdo it. Have fun, and best wishes -- this is such an exciting time!!