Is there any place in AustinTX where I can play mud and be easy to clean and not disturbed by passersby?

Based on the criteria you have specified, definitely not. You might from time to time find some mud on some of the lakes when the water level is down that meets your requirements...

Which lakes?

Lake Buchanan and Lake Travis...

Thanks

These two are quite large areas, do you know the specific places?

As you've already discovered on Lake Travis, the mudflats on a lake are usually upstream at the top of the lake or in a bay that is drying out. The only way to know where mud is or isn't on a lake for sure is to go look and checking a lake out for mud is easiest to do with some sort of boat...