There are several different affordable rental housing programs.
The U.S. Department of Housing and Urban Development (HUD) funds:
Public Housing, which is owned and operated by Public Housing Authorities
Project-based Section 8 and other subsidies directly at apartment complexes
Section 8 Housing Choice Voucher Program
You can search for a Public Housing Authority at this link: http://portal.hud.gov/hudportal/HUD?src=/program_offices/public_indian_housing/pha/contacts/tx
You can search for a HUD-assisted apartment complex at this link: http://www.hud.gov/apps/section8/index.cfm
You can find open Section 8 Housing Choice Voucher Waiting Lists at this link: http://affordablehousingonline.com/open-section-8-waiting-lists
There are also apartment complexes with somewhat affordable rents through the Tax Credit Program. Tax credit landlords are not allowed to discriminate against Section 8 voucher holders. You can find tax credit properties at this link: http://hrc-ic.tdhca.state.tx.us/hrc/VacancyClearinghouseSearch.m
You can find a list of licensed boarding homes in Dallas at this link: http://boardinghome.org/licensed-homes/
You can find a list of boarding homes in Fort Worth and the mid-cities at this link: https://www.yellowpages.com/fort-worth-tx/boarding-houses These may or may not be licensed.
You can find a list of licensed boarding homes in San Antonio at this link: http://www.sanantonio.gov/Boarding-Homes-Ordinance
You can find a list of “personal care” homes in Houston at this link: https://hcpc.uth.edu/pages/help/local-care-homes.htm