Well, I'm going to give you more information than you probably want.  Fortunately, I'm definitely not including photos of these ugly pests!

According to WSU, there are three insects that attack blackberries: 1. Raspberry Crown Borer: found in canes; 2. Raspberry Mite: a tiny (1/50") sausage-shaped yellow mite found in berries; and 3. Spotted wing Drosophilia (SWD), found in the fruit.

Without seeing the infected fruit I can't say for certain, but most likely from your description, I'd bet it's Spotted wing Drosophilia (SWD).

Here's some general info to help identify the pest according to  WSU Hortsense:  "This exotic pest has recently been found in California, Oregon, Washington, and Florida, as well as in Hawaii. It resembles other Drosophila species (fruit flies or vinegar flies) in appearance, but unlike other members of the family which attack only overripe, damaged or decaying fruit, SWD attacks healthy fruit as it ripens on the plant. Adult SWD flies are about 1/8 inch long, with red eyes and a yellow-brown body. . . . Eggs are laid beneath the skin of ripening fruit."  

To identify:  "Scars left by oviposition (process of laying eggs) may appear as indented, soft spots on the fruit surface.  Small white- or cream colored larvae hatch within a few days and feed in the fruit, causing the fruit to soften and collapse around the feeding sifte.  At maturity, the larvae may be up to 1/8 inch long.  They may pupate inside or outside the fruit."

WSU and other universities are actively studying ways to control SWD, especially for commercial growers.  WSU Master Gardeners state, "SWD is considered a serious threat to fruit and berry crops including cherry, peach, nectarine, plum, plout, thin-skinned grape, raspberry, blueberry, and strawberry" and more.  Yes, another pest to worry about!

So, what to do if you see berries with SWD?  Do not compost it. Instead, destroy infested fruit by burying it 18" deep, crushing, or bagging and discarding in the garbage.  If you have any bushes in your yard, the above links have advice for controlling the pest in backyard gardens.

Although I'm giving you lots of info, I would not stop picking berries.  Just check them over carefully and if you see something suspicious, don't add it to the pie!