The first step is to find out what are the legal means of controlling them in your state.
Here is a link to some information for Missouri that I found useful.
https://mdc.mo.gov/wildlife/nuisance-problem-species/nuisance-native-species/armadillo-controlI have an armadillo infestation on my land. The only thing I have found useful is the injection of a small amount of lead at high velocity.
They will be very difficult to trap (if even legal), but they sometimes do have obvious travel trails on grass. They are also very difficult to poison, since their preferred diet is live insects and insect grubs.
I have tried castor oil around my new tree plantings. Supposedly it makes the bugs and grubs taste nasty and they will not root around that location. It would cost way too much to spray your entire dam several times per year.
Sorry to not be of more help - there is no easy solution. I gave my nephews a bounty of $5/armadillo. So far I am out $20.
If your dam is in grass, I might try burning it during the proper season. That should at least eliminate the insects for part of the year and perhaps the armadillos will then hunt other ground.
Good luck on your battle.