It may sound odd, but you CAN do the day there even if not a strong swimmer.
You don't have to swim to do the 'swim with the dolphins,' as the deep water part is very, very limited (just a quick dorsal pull back to the wading area) and you're welcome to opt out of that part.
They also offer nice lifevests, which provide a ton of buoyancy and will keep anybody afloat. When I worked there, we'd see tons of guests (lots of the British ones, in particular) using those to float across the deeper water areas in the River and in the Reef. There's lifeguards everywhere, so no worries even if you have a problem.
The Aviary is accessible by both land and water, so weak swimmers can walk to it. The Stingray Lagoon is a wading exhibit (around three feet or so deep). And they offer complimentary snorkeling lessons if you need assistance, too.
So, strictly speaking, you don't have to be a strong swimmer to do a day at DC. =)
And also... shout out to my home state! I was born in Montgomery, but most of my family lives around Lake Martin or in Auburn. =)