Audio drift can have a couple of causes, dropped frames (which a TBC can help with), and unstable clocks, where the audio isn't recorded at quite the precise rate it should be, causing drift, as Michael correctly pointed out.
Dropped frames caused by tape dropouts are a problem if the device skips a frame, without skipping the corresponding frame of audio. A TBC helps with this, as it will ensure the vertical sync pulse is there for each frame, even if the tape has quality issues - it will ensure your capture device always sees a frame. LOTS of older PC capture devices I tried were impacted by dropped frames. The most reliable method I found was using a dedicated HDD/DVD recorder unit, all the ones I used would always maintain the correct frame rate (without input via TBC), and the audio would always be in sync. I also sometimes used a MiniDV deck to achieve the same, which has no choice but to always record a frame, and therefore keep audio/video in sync.
Doing it via PC, I've only had success with 1 device, a "Dazzle DVD Recorder HD" that I bought in 2011. It came bundled with a copy of Pinnacle Studio HD, and it all runs on an old Windows XP machine I keep for that purpose. I don't know if it is the hardware or the software (or maybe the combination), but this device has worked for me in transferring tapes and keeping things in sync, and I've not had to buy another device since coming across this one.
If you can find a combo VHS/DVD recorder unit (I'm pretty sure they are no longer manufactured), they are usually the most reliable solution to doing this without a lot of effort. A DVD Recorder (with analog inputs) will also do the job, mine had a HDD which you would record to first, and then output those files onto a DVD. I stopped using the unit when the DVD drive eventually failed. Otherwise you are stuck just trying the various analog USB capture devices on the market, which like I've said, I've found to have mixed results, and only come across one device so far that does the job reliably.