Background:

My partner and I rent, not own, and we live in central Texas. We are originally from the city and know nothing whatsoever about lawn care. Unsure what kind of grass we have, but the area in question gets full sun.

Issue:

We started renting our house last winter and inherited a pretty bad lawn, which we didn't realize for awhile since the plants were dormant for the year when we moved. Had no clue spring was going to rain hellfire down on our lawn until it was too late. Within two weeks, the yard was a total mess; probably 80% of the plants are variations of weeds, and I'd say the grass by our driveway is maybe 90% crabgrass in particular. It goes deep; the previous tenants had dead/dormant crabgrass already there when we moved in (wish I'd realized what it was at the time so I could have removed it then), and that spawned a huge problem this spring.

I'm having a hard time figuring out how to fix this lawn. The crabgrass is so hard to dig out, and once I manage to remove it from an area, I'm pretty much left with just dirt there. I don't know if I'm even removing it the right way (hand pulling, trying to use a shovel on the tough ones), or what to do once I've removed it. I've tried looking at articles and videos on how to deal with crabgrass, and most of them seem to assume you don't have very much of it, or it's early enough in the year that you can just lay down some preemptive weed killer and you'll mostly be fine. It's May now, and most of the info I've read warns that you should not reseed in the spring either, so I'm not sure what to do once it's all out and my yard is mostly dirt (we have an HOA, I don't think I could get away with an empty lawn until fall).

Also, we're renting, so I don't know that we can do anything drastic, like pull up the whole thing. I certainly don't feel inclined to do anything expensive or anything that's going to need several seasons to follow through on, since we'll probably be gone by then. I just want a nice enough yard to not have the HOA and our neighbors angry with us for being that house.

Any suggestions would be so, so appreciated, I really do not know what I am doing here.