Can I run TP4056 modules from the same DC power source?
No, you can't use the module to charge batteries in a same pack simultaneously with the same power supply. That will short out the batteries via the modules together.
The modules are meant to chage one cell only. Connecting them in parallel means shorting the battery cells as they are connected in series inside the pack but parallel at the poer supply side.
And that is the reason why your modules burned up, and that is the reason why other people use isolated power sources to not burn anything up.