You should send the owner a certified letter demanding receipts, and indicating how much money you have paid already. In Texas, a landlord is required to provide a receipt if the rent is paid in cash. See Section 92.011 of the Texas Property Code. A copy is here:
A landlord who fails to follow the law can be easily sued in justice court. Hopefully it will not come to that. If you want receipts for money orders or checks, that is more difficult. Money orders can in theory be traced, but it is expensive, a hassle and takes time. Sometimes, the trace does not show anyone has cashed the money order, so you still do not have proof of payment. Thus, I agree that the landlord should give you a receipt for a money order too. You might try to see if the landlord will initial or sign your money order receipt that the store gives you (or sign the carbon copy of the money order that some have). You could also create your own receipts. A landlord or manager that refuses to provide receipts or sign your receipt you create cannot be trusted, so beware. You may be in for trouble. You may want to contact the owner of the property because the manager might be planning to steal from you and the owner soon (sometimes managers do not tell owners that they have even rented out certain units and are just pocketing the money).
I would be nice, but insist that you get receipts for all payments. If that does not work and you don't want to take more aggressive action, take a friend with you when you make the payments at least (and consider moving when your lease is up).