How to adjust the offset of a full-wave precision rectifier?
I'm afraid you've missed the point of the circuit. The second op amp does not need any diodes. You can adjust the offset with one adjustment of the second op amp.

There is one slight issue which you might not have considered, though. Unless R8 and R9 are exactly of a 1:2 ratio, the output will have slightly different gains for signals of different polarity. Since you appear to be looking at overall amplitude of an AC signal, this probably is not a concern.