Backyard makeover suggestions (possible conversion to Bermuda)

Moved into a new home in late 2022. The backyard is a mix of St. Aug and Zoysia. The rear of the backyard slopes towards the house.

I am having issues with my ground being incredibly soggy for longer than it should be. Curious if the group thinks French Drains are the solution or better grade work towards the side of the house and towards the street.

On to the grass. My backyard faces East with one piece of shite tree in it currently. Surrounding trees from neighbors pose little to no issues with nearly full sun until the sun obviously gets on the other side of the house. The least amount of sun in the backyard might be the strip along the S. Side fence.

The Zoysia does mostly OK, but caught a bad run of bugs/disease late last year. It really thinned out. The St. Aug has become straggly and lots of exposed runners. This is probably attributed to the extended moisture captured in the ground.

So what does the group think are my best options:

1) better drainage and revive the St. Aug & Zoysia

2) Nuke the grass, have someone come do dirt work grading, and go with Celebration Bermuda

I have two young kids and two big, girl dogs. Dog piss and traffic are also contenders in threatening my grass. No irrigation in the backyard but I do water pretty diligently. I can take pics at lunch, if that helps too.