Shedd is big enough that it's probably best to just use the restrooms near you. I think the best restrooms are the ones in the entrance lobby (though on busy days, these are hard to reach) and those by the Bubble Net Cafe (middle level on one side of the Oceanarium). The Wild Reef restrooms are pretty good as well. The lower level of the Oceanarium has nice restrooms, but often a long wait (the only restrooms are two family restrooms).
The only restrooms I found with paper towels were the ones in the Wild Reef. All others had modern (and loud) hand dryers. This is not surprising for an institution focused on environmental impact, but can still be problematic for young children or children with sensory challenges. All toilets are fairly typical (no smaller preschool toilets, for example).
Entrance Lobby
The best restrooms at Shedd are in the main entrance lobby, where you buy your tickets. When you come in, along the right side, are signs for lockers and restrooms—follow those. There are men’s and women’s restrooms, with nice changing tables and lots of stalls. In addition, there is a separate family restroom with an adult changing table, and a separate nursing room which is essentially a family restroom plus a comfortable chair for nursing. The non-family restrooms are long and narrow, but you can get a single-wide stroller in without too much trouble, and there’s space to park them at the end by the changing table. Double-wide strollers will be difficult, and there’s unfortunately not good space outside the restroom to leave them.
Wild Reef
The men’s restroom, along with a family restroom, is in an alcove just before the large room with sharks and rays, and the women’s restroom is in the alcove just after the room, along with another family restroom. All four restrooms have typical fold-down changing tables and have paper towel in addition to a hand dryer. There is some space for strollers in the general restrooms, and space to leave strollers outside as well.
Oceanarium - Top Level
The top level of the oceanarium (with Soundings Café, the 4-D movies, and the entrance to the aquatic show) has the worst large bathrooms in Shedd. The bathrooms are located along the walkways between the Caribbean Reef area and the oceanarium—both walkways have similar bathrooms. The bathrooms are long and narrow and often crowded, and it would be difficult to get even a single stroller through the length of the restroom if there are others inside—there is, at least, room to leave strollers outside easily. On both sides, the only changing table in the men’s room is inside the accessible stall.
Oceanarium - Middle level by the Bubble Net Café
The restrooms by the Bubble Net Café are pretty good. They’re large, though at top eating times they can get moderately busy. Both restrooms have typical fold-down changing tables.
Oceanarium - Bottom Level (underwater viewing)
At the underwater viewing level, bathrooms are by the “Shallow Ocean” touch area. Look for the small hallway by the stairs. The hallway leads to the elevators and two well-sized family restrooms, with fold-down changing tables.
By the submarine play area, there are signs for a restroom, but they end up just pointing you to the elevator to go up to the not-great restrooms at the top level of the Oceanarium.