Answer: I think you are dealing with two different problems here - squash vine borers and incomplete pollination. I can't make a positive identification but the damage you're finding sounds suspiciously like the work of squash vine borers. These bugs like to hide in leaf litter or under boards, stones, etc. at night. Since they overwinter under garden debris, and there is only one generation per year, it is possible to reduce their population with diligence early in the season. You can place boards around plants, then go out early in the morning to kill any bugs hiding underneath. Also, search for and squish their shiny yellow to reddish eggs (on the undersides of the leaves), as well as any nymphs you find. Pull mulch away from the base of plants too, to eliminate hiding places. Row covers can exclude the insects, if placed on plants early in the season. You'll need to hand-pollinate flowers, however. There are also some resistant varieties of summer and winter squash so try planting those next year.
As for the rotting squash, it can be a disease called blossom end rot or it can simply be poor pollination. That's when a fruit begins to form but the vine aborts it because it was not pollinated and will never mature.
You can replant summer squash now and still get a harvest, but I'd suggest planting in a totally different garden bed until you can get the insect and disease problems under control.
Best wishes with your garden!
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