An acorn tree? Do you mean oak? And is it a live oak (little leaves) or an eastern oak (huge leaves)? In the pic it doesn't look like either one.

How long to wait until using weed n feed again? 10,000 years. You may have killed off your remaining St Augustine with that product. We don't like wnf products for several reasons. In the south you can add the problem that it kills St Augustine.

Bermuda will not grow in that shade. St Augustine should thrive in that shade...if it doesn't get too much water. This has not been a good year for bermuda because there has not been enough sunlight. In your neighborhood it has not been a great year for St Aug, either, because of too much water. I have a fungal disease in my St Aug simply because of the rain.

Your St Aug looks fine in the pictures. If you get any dead or even thin spots forming, the solution in Texas is corn meal. Ordinary corn meal (not corn GLUTEN meal) attracts a predatory fungus that wipes out the disease on the lawn. It takes 3 weeks to work, but it works. So keep an eye on it. The disease will not go away by itself.

St Augustine is the only common turf grass that will thrive both in your sun and under your tree. There are a few others, but those will cost you. If you do nothing but raise your mower to the highest notch and mow every 2 weeks, that will take out the bermuda and most other grassy weeds. It takes time as the St Aug sends runners out into the bermuda, but as it sends more and more out, you'll have less and less bermuda. I'm using that approach on my new house. I had about 10 square feet of St Augustine last July. Now I have about triple that. It'll get there.

About watering. Once the rain goes away you need to wean the grass off of the constant moisture and start withholding the irrigation. Your rectangular yard is perfect for an oscillator type sprinkler on your hose. Get a good turbo model for about $20. Then put out some cat food or tuna cans in the yard and turn on the sprinkler. Time how long it takes to fill the cans. That will be your normal time to water from now on. My oscillator takes 8 hours to fill the cans. Every system is different, so you have to time yours. That time represents 1 inch of water. Then the secret to make this work is you only water once per week in the hottest heat of summer. If it turns out you have a summer with many days to weeks of temps in the 100s, then you can go to once every 5 days. But NEVER water every day or every other day. My buddy in Phoenix (temps running in the 110s) waters his St Augustine on Fridays and Mondays before sunrise. If he can do it, you can do it. Here's a picture of his from yesterday...

This is looking southwest in early morning, so that distant part of the lawn gets full shade all afternoon.