How best to fill the spaces between flagstone?

I would use sand or small stones, like pea gravel.

Polymeric sand is what I would use.

https://www.thespruce.com/what-is-polymeric-sand-2132510

If you prefer something to grow between the flagstones, check with a local greenhouse nursery, not a big box store, and see if they may have a ground cover that would grow in the shade.

Here's a great video for you. The ratio is 2 part cement to 1 part sand and just enough water to make it into a crumbly mix. You can even add a mortar color into the mix. Good luck! https://www.youtube.com/watch?v=dRC0nyisC5U

If you do not want to pull all the flagstone up and redo it, just do a little at a time by digging out and removing the dirt and refilling the area with sand or mix up small batches of Sakcrete concrete and fill in the areas where you've removed the dirt. The Sakcrete will dry and make the entire area more stable.

Just dig out the soil from in between the flagstone down about two inches. A shop Vac would help. Then just use mortar mix to fill in.

Polymeric sand for sure!