Transferring directly from tape to a (good) DVD recorder is the easiest path to a digital copy, but you pay for it later if you decide you want smaller or more universal files. "DVD" is basically a dead end format: perfect if you will be happy only viewing an actual dvd disc, but a nightmare if you also want to view the videos on a phone, tablet, netbook, etc. "Re-purposing" dvd video can be difficult.
Converting the dvd data into more portable formats like MP4 is possible, but tends to be plagued with glitches like the audio drift you mentioned. Other issues can include steep video quality decline and spurious added artifacts like bizarre interlacing or "washboarding" that wasn't there previously on the original dvd. Simple conversion utilities are most prone to this, more complex software with command-line programming is better but has a learning curve.
If you don't mind large files, the most reliable direct conversion utility is
MakeMKV. This quickly rips the VOBs from your disc and re-packages them as standard MKV video files. The original audio/video is not altered at all, just wrapped in an MKV file container. Most modern devices and computer OS can play MKV, its not quite as universally playable as MP4 but almost. The drawback with this workflow is the resulting MKV files will be as large as the original VOBs: up to 4.3 Gb if the entire dvd consists of one single recording. If your dvd has been edited into several titles (menu items), MakeMKV is a bit more practical (it creates an individual smaller MKV file for each title). But these will still be larger than usually desired for posting to social media, emailing, storing on a phone, etc. Employing
Handbrake (as hech54 mentioned) on the MakeMKV files can reduce the size, depending on your computer OS it may also be possible for Handbrake to bypass MakeMKV and convert directly from your physical DVD discs to smaller MKV (universal) or M4V (
iTunes) files.
If your primary end goal is MP4, you may need to re-think using the DVD recorder at all. It might be best to re-do the project from scratch using standard direct-to-PC capture workflow, described in many threads here on VH. Many later conversion problems are avoided by not using DVD VOB/MPEG files to begin with. Of course, direct PC capture is not as easy as using a dvd recorder: some people never get that to work well for them either. DIY tape digitizing often leads to "a rock or a hard place" decisions.