Thickeners are generally used in silk painting… and by painting I mean the kind of painting you do when you want to achieve a picture. I wouldn’t use it for yarn dyeing, it seems a bit of a waste. If you want to avoid puddling, be a little more stingy with the dye, be a little more careful in how you apply it and elevate the surrounding sections that you’re not dyeing (put them on a plate or in a bowl or something). Dye one color at a time, let the yarn soak up the dye then lift it and move it and clean up the mess underneath before starting with the next color.
You can also use a strip of cloth (I’ve used muslin) to tightly wrap either end of the section you’re working on to help prevent it from pulling up the dye.
Another thing you can do is only put colors next to each other that will achieve pleasing results… red next to yellow will give you orange, blue next to red will give you purple, etc.