Attorney here, but I'm not giving you legal advice, blah blah disclaimer blah blah.
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The concept of mitigating damages is a general concept that applies into every of area of law, not just landlord/tenant. Basically, you can't sit on your ass and/or let things get worse, and then claim that the other party should pay you for sitting on your ass and/or letting things get worse.
An important point, though, is that the person who has a duty to mitigate only has a "reasonable" duty to mitigate. They don't have to go out and make it their life mission to mitigate; they just have to make some reasonable effort to mitigate their loss.
To illustrate -- I had a construction case where the plaintiff sued the builder (my client) for alleged construction defects, then didn't put a roof on the property over the winter (Ohio). Then the plaintiff wanted paid for all the damaged wood. You guessed it -- court told him to go fly a kite, because that's completely unreasonable.
But here, it seems to me your landlord took reasonable efforts to mitigate. The place was only vacant for six weeks. That's pretty damn good (think about it -- the landlord also needs a tenant to be ready to move on from his/her old lease). I don't think there's a mitigation issue at all.
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As for the security deposit issue, I'm not going to spend the time to research that in Texas. I'll defer to other people here, and it seems like you should get that back by sending notice to the landlord.
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But as far as the other stuff, it seems to me that you came out ahead. I had to terminate a lease in Cleveland and had a much, much, much stiffer buyout than you.
Think about the practical matters, too. If you file a small claims case, you are going to have to take time off work to file the complaint, to maybe attend a pre-trial hearing, to maybe respond to discovery, and to appear at trial. You want to use vacation time for that?
And trust me, if you don't have a lawyer, you're probably going to go into small claims and look like a doofus. Even the most educated people in there who think they know what they're doing end up making a fool of themselves, especially if the other side has a lawyer. Here in Ohio, a corporation must be represented by counsel at any hearing; so if the property is owned by an LLC, and Texas law is similar, your landlord will have counsel. Which means you should have counsel. Which is expensive.
You want to do that over this amount of money?
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What I'm getting at here is if you came into my office, I would tell you to move on, and that this fight isn't worth it. But maybe your priorities are different, and if so, go have at it. Best of luck to you.