I can suggest several options, Sloan. One is to simply buy a pair of all-purpose “water shoes.” Something like the Keen McKenzie shoes ($90). These are like a low-cut trail-runners or light hikers, but made from water-resistant materials and designed with mesh panels that allow water to drain away. They also have grippy rubber soles and front bumpers to protect your toes.
Alternatively, you could investigate purpose-made shoes for wet wading. Muck’s Cikana Fishing Boots ($120) are basically lightweight rubber boots that fit above ankle. They've got flexible cuffs to keep water out, and fine-tread soles that almost resemble fish scales and offer excellent grip on smooth rocks. You might get some leakage when standing in water that runs over the boots, but in shallows you’ll be fine.
Interestingly, felt-soled shoes, long popular among fly fishers, are falling out of favor and are even illegal in some states. That’s because the felt tends to catch and spread invasive algae. Several companies, Simms for example, stopped making them altogether (Simms has since begun to produce felt-sole shoes again). Instead, they offer a line of shoes that use sticky rubber and fine-pattern treads (like Muck) to create shoes that will keep you upright on slippery rocks. Simms’s Rivershed is a good example in an above-the-ankle light wader made from rubber and synthetic leather. Inside, neoprene liners keep you comfortable when wet. They’re a bit pricey at $179, but nice shoes.