Hi and welcome! I always grew raspberries until I moved to Florida, and the one rule that my Dad taught me (I still remember clearly!) is "good foliage makes good tasting berries".
So, first thing is to keep the foliage healthy until after the berries are ripened. Enough water, and fertilizer going through the summer are important. For fall bearing raspberries, I'd be giving them an extra top dressing of fert and compost now to carry them through. Be sure to water extra if you have hot, dry weather into September as the berries can't size up without enough water. Be sure to keep the weeds pulled out from amongst the canes so they aren't competing for the goodies. Mulch is helpful around the bases of the canes to discourage weeds.
The other thing that could cause the later berries to shrivel up is bad weather. If you get a run of cloudy or rainy weather just when the last berries are coming along, there might not be enough sun for the plants to make good fruit. Even good leaves can't make sugar without sun. The only answer to that is to interplant your old bushes with another variety that bears earlier, or an "everbearing" type that will give you a lot of fruit in June/July and then keep bearing.
