As you have discovered...autumn fruiting ones don't die down same as summer ones. You have now a options with the cutting down..
1..cut all 'old' canes down and have the usual late crop of berries..
2. ...thin out some of the 'old' canes, and those you leave will produce for you 'summer crop'..once the first crop come to an end, cut the 'old' canes off to make room for the new canes for the usual 'autumn crop'.
If you decide to double drop you raspberries...remember to give them some feed when you clear last of the 'old' canes off. They will need some 'snap' for working so hard for you. It doesn't 'damage' or harm the future cropping/growth. I used to do this for years as regular 'habit' and as long as I kept the plants fed and gave them some water too when it was really hot, they just kept going on producing plenty of tasty berries
There is small waiting gap between early and late crops though..it won't be 'non-stop' fruiting...