Do they have a clinic for low-cost annual vaccinations and/or senior citizens pet care discounts? Walk-ins or appointments?

Being as busy as they are, you may want to check out their website https://dallascityhall.com/departments/dallas-animal-services/Pages/default.aspx or facebook page.

I recently been there they offer free vaccinations, neuter and spayed your animals lets not forget chip them also on Tuesdays starting at 8 am. Still you can call and inquire more info. It is also free if you stay in the South Dallas area check and see if your zip code qualifies

Discounts no but low cost yes. And you can call up there to see when the next one is.

I would call

I would make sure and call