Browning's stock finish is notoriously hard to remove and to sand.
320 grit is very fine, much too fine for initial sanding after stripping. 600 grit is finer than the grain of the wood. Start with 220 grit, move up to 320 grit, then 400. If you want it smoother, sand the finish lightly between coats with 400 grit. Do not use steel wool at all.