After that, it will become a dance of either having tenants stay and effect small increases annually or lose tenants and then correct at once to market.  

But property prices are increasing so the rents but ratio .5 or so  percent remain same. 

Imagine the increase in your holding costs while your property sits without selling, take that money and put it into fixing this.

The problem now is trying to make those big rent increases.

Any large bank is going to do commercial loans for their customers.

My recollection is that if it is your primary home for 2 out of the last 5 years there is no capital gains tax on the increase in value up to a $250K profit for a single person or $500K for a married couple. 

inherited" tenants like yours are probably waiting to hear from you, so send out contact emergency info and have them respond and include with next rent payment a updated contact sheet,, I'd make an appointment to tour inside of each unit with condition sheet,, the condition from your date of ownership is now what you use for condition move out.. so once that's completed you now know which unit would be tops on your list to rehab and offer an early out to.. if that's a consideration.. you can of course let all tenants know the rent will increase to $$$ when their lease is up and you are accepting 30 or 60 day vacate notices early, they'll be prepared. 

It has been many years but tax is not increasing much. 

Make friends increase your circle of influence - Hire a private detective - they are good at finding people (give his a piece of the profits)Check the death notices or search government record for certificates of death, check the obituary for next of kin - contact them. 

That being said, the purchase price is below $100 per square foot and I am betting the steady flow of increasing population to come to the Houston area as it has for the past several years.