I am very sorry that you are experiencing this issue. Typically the audio-video sync problem can occur for either of the below cases:
1. Your system is low on resources during recording. If this is true, you would be able to notice the audio-video lag right after recording (i.e., while previewing the recording, working with it on the timeline, etc. before producing).
2. You are producing a project with full-motion video from a digital camera at a non-standard frame rate. You can find the solution at: http://techsmith.custhelp.com/app/ans...
One final workaround that you can try is producing to the AVI format using Custom Production Settings path. If the lag is still noticeable in the produced AVI file, then the problem was likely caused by (1) above. If it is not, you can re-import the AVI into a new project in Camtasia and try re-producing to the desired format.