What is the best way to turn soup into stew without using flour?
Let's leave aside the question of what separates a soup from a stew (there's no real answer, only mostly arbitrary opinions - which seems to be a somewhat widely shared belief around here: https://cooking.stackexchange.com/a/20963/70120).  It sounds to me like you have a dish with some liquid in it and you want to thicken it.
There are a number of ways that soups/stews are traditionally thickened.  I'd break them into four or six categories depending on how you want to count:

Add starchy paste: this is the method you've already found, adding slurry made from flour and water.  If you have a problem with wheat in particular, you could use corn starch, potato starch, etc. instead.

Add a starchy ingredient and cook it: a similar idea, but rather than adding a slurry of flour/starch and water, you add some starchy food like noodles, bread, or potatoes.  If you cook these starchy foods in the liquid for long enough, the starch will leech out of the potato/noodle/bread and into the liquid, having much the same effect as the first method. You can speed up this process by adding smaller particles, for example, shredded potatoes, or bread crumbs.

Add gelatin: this is not much different than adding starch, except that you usually don't need to make a slurry, you can add the gelatin directly and it may have a lesser effect on the flavor.  I'm including in this category thickeners like xanthan gum, or pectin, or other vegan alternatives to gelatin.

Add a gelatin-containing ingredient and cook it: rather than adding gelatin directly, you can add an ingredient with gelatin in it.  The most obvious choice here is bones. There's collagen in animal bones, as well as in skin and connective tissue (like you'd find in a tough cut of meat) and when you boil them (especially in something acidic) you'll get gelatin.

Add cream: cream itself isn't much thicker than water, but the fat in the cream can make the soup/stew feel, well, creamier and less watery.  Other milky ingredients can be used, including thicker ones like sour cream or yogurt. Generally speaking, the thicker the milky ingredient, the thicker the soup/stew, but if you add sour cream and then cook for a long time, it probably won't seem much thicker than it would have with milk. When we add thick milk products like sour cream and yogurt, we typically add them right at the end.

Blend the soup: rather than adding a thick or thickening ingredient, you can add structure to the liquid by blending in whatever you were cooking in the liquid.  This is very common in French vegetable soups.  Often only a portion of the soup is blended, so that you end up with a thicker liquid and some large pieces of vegetables and meat.