Your goal is to increase water clarity and decrease filamentous algal growth. I assume that your clarity is limited by algal bloom most of the time and not by suspended clay but this is easily verified.

You have 20 years of nutrient loading in very shallow water. Dredging will reset the clock on that.

Aluminum sulfate treatment will reduce soluble phosphorous (needed for algae/plant growth) temporarily. You could also look into adding pond dye to block photosynthesis though some people don't like the Tid-e-bowl color. Tilapia will eat algae, but they also die off en masse when the water cools and you won't have largemouth bass in there to clean them up.

Water testing can be done on the cheap through Texas A&M by mail.

http://soiltesting.tamu.edu/files/waterweb1.pdf