Texas: running multiple apps at the same time?

[Elizabeth Goff]

I just found out the property manager of a duplex I own in Texas is only running applications concurrently.... so if the first one fails/drops out, then they start on the next one... which obviously can extend the time to get someone placed.  This is not how my other property managers (or I) process applications, albeit in different states.  I run at least two that look good on paper and choose the best.  She is telling me that that is against the law and/or code of ethics to charge app fee and process more than one at a time in Texas as you are promising/selling the same unit to multiple people.  Doesn't sound right to me, but wanted to get feedback from those of you who know the rules in Texas!  Thanks in advance.