Damar is not completely soluble in mineral sprits, but requires turpentine or something stronger. Oil paint doesn't need damar in it anyway. As for your brushes, dishwashing "soap" is not soap, but detergent, and detergents are too harsh to be used on natural-hair artists' brushes. Soap and lukewarm water are preferable to detergent. Try suspending the ones you mentioned in safflower oil for a few days (don't let the bristles touch the bottom of the container or they might bend and stay bent) and then wipe them off on a soft rag and wash them gently with soap and mildly warm water. Hang them to dry with the bristles pointing down, to keep the water away from the wooden handles. If they are still splayed out, they will be good for painting things like grass, moustaches, etc., at least, and then you will know to treat your new brushes better, and why you should.
Remember that these are made of hair. What would the hair on your head do if you were to use dishwashing detergent on it? It would frizz out like crazy, break off, get split ends, etc. Not what you want the hair in your brushes to do. The oil acts as a conditioner. I poke small holes in the plastic lid of a coffee can, and stick the brush handles through from the underside far enough that with the lid on the can, the bristles of the brushes are 1/4 inch or so above the bottom of the can. There is an inch-and-a-half of safflower oil in the can. I can put the brushes into this safflower oil with paint still on them, and the paint will stay wet until I get around to using the brushes again or cleaning them as described above.
Virgil Elliott