Tenant Gave Notice AND Didn't Pay Rent

[Marco G.]

I know a lot of tenants think their deposit can be used for last month's rent, but this isn't one of those situations.  Tenant is a nurse who always paid on the first, no exceptions.  She was always trying to get rent reductions (i.e. when the AC blew and it took two days to do a complete new HVAC system) and I believe her decision to vacate is based on the fact that when she discovered her glass balcony door was shattered when she got home at 8pm that it took until the next day to get it fixed.

Regardless, her deposit is less than one month's rent, and she has't paid.  A 3-day notice to pay or quit was sent out on the 6th, and still nothing.  I don't doubt she'll leave, but if she does without paying, what's my recourse?  Or does it make sense to start eviction proceedings and get her served before she leaves?  Would I be able to get a judgement even if she has left?

This is in Dallas, TX.

Thanks in advance for any insights!