Instead of spending that much money on a new coop, what I would do is 1. Take the coop you already have and modify it to be a nicer home by taking out the floor, covering the sides down to the ground, putting in a roost about 18" from the ground and putting in hardware-cloth panels for ventilation a foot above the roost. 2. Then use the $250 in your budget to buy a chainlink dog kennel, enough poultry wire to cover the walls and top, and a roll of 4 foot tall hardware cloth. 3. Install the hardware cloth so it goes around the bottom of the dog kennel two feet high and bend to extend along the ground another two feet out to form an apron.
A bigger run will allow your birds to find more protein for themselves, like bugs. Supplementing more protein such as mealworms, and more calcium in the form of oyster shell, is a great plan.
Though feather-plucking is a bad habit, I don't think you necessarily have to cull your hens before trying some different things. I once had a really bad feather-plucker, and isolating her from the other chickens for a month broke her of that habit. Putting some interesting things in their run might help, such as piles of rocks to hide among, rotting stumps to jump on and peck at for bugs, maybe a chicken swing? Stuff to reduce boredom and give them other things to do.
Of course, this is after you rule out lice and mites.
Good luck, I hope you find the right combination of solutions.