Can I use limestone screening to fill paver joints?

Limestone screenings eventually become acidic when wet and can eat away at pavers eventually. Your grass also won't like it.

Did you buy good polymeric sand? And did you install it/mist it properly? Because thats what I would recommend.

How large are the gaps?

go with techniseal polymetric sand.
If you want better/stronger, go with their commercial line.....

Maybe I had a problem compacting the polymeric sand.
The gaps are 1/4 inch big.