CARES Act puts a moratorium on landlords sending a notice of eviction for certain "covered properties" until July 24th. So even though Texas allows evictions, if the landlord has a "covered property," the notice of eviction can't go out until July.

What's a "covered property?"

"The CARES Act’s eviction protections only apply to “covered dwellings,” which are rental units in properties: (1) that participate in federal assistance programs, (2) are subject to a “federally backed mortgage loan,” or (3) are subject to a “federally backed multifamily mortgage loan.” "

https://crsreports.congress.gov/product/pdf/IN/IN11320

How do you know if you live in a "covered property"? Ask. Some landlords and/or property managers might not even know about the extended moratorium. Most JP courts are requiring landlords file an affidavit that this isn't a covered property.