Looks
very similar to my 1960 Whirlpool, if you find a source for any parts, please let us know.
The Bleach should dispense at 4 minutes left in the wash agitation of most (maybe all) cycles. The softener should release exactly when the final deep rinse water starts entering the tub, as the tub slows down from spinning when the extraction is done.
Whether or not you get both sides working, for any load where you won't bleach or won't soften, pour in 3/4 cup of warm water. It helps keep the rubber stopper in good shape, helps clean out a less-than=perfect dispenser design.
Very nice washer, congrats!