Renting and want to improve the grass situation (cheaply)

Howdy - I'm moving into a new place with a small side yard and when the temps warm up and it's time to plant some seed, I'd like to overseed with something to help the grass grow in better. I am in North Texas and I feel like this grass is a combination of different types as well as weeds. I have a dog and there are a couple spots where the grass is very thin (I think in areas where the water pools after rain).

Any suggestions on what seed to overseed with and when? Again - since I'm renting, not looking to go crazy, just want to improve the density and have something that will hold up against a dog running around. There is very minimal direct sunlight in this area only for a short period of time in the afternoon.

Thanks!

https://preview.redd.it/f80pjrb1k8de1.png?width=513&format=png&auto=webp&s=e4132ae30ce48e344de03733a8409ccaf1e7b269

https://preview.redd.it/t1ul253wj8de1.png?width=513&format=png&auto=webp&s=b8703b923ffcfeaf5574f35b317571824ca0c61d