Camera with native recording resolution higher than DVD, will the recording look worse?

You ask a really complicated question.

First DVD is 720x480i (not p) and anamorphic (pixels are not square). There's a flag which determines if the encoded video is played at 4:3 or 16:9, but in practice I've never seen a DVD which uses it (at least not in the U.S.) and they're all coded as 4:3 with non-square pixels.

PAL DVDs are 720x576i, but at 25 fps instead of 29.97 fps, so you gain a little resolution but lose timeframes.

From what I can dig up on Google, the DXC-637 records in 800x400 or 800x450 resolution, presumably interlaced.

So yes the image will look very slightly worse after converting to DVD. First because of the lower horizontal resolution of DVD. Second because it needs to be resampled for the anamorphic rectangular pixels and up-ressed to 480 vertical lines. In practice, the difference is so slight that I suspect your conversion hardware (are you using composite cables or s-video?) will have a bigger impact on video quality.

If you want to retain maximum quality, I suggest finding some way to connect the camera up to a computer, and capturing the video into a modern format (MPEG2, MPEG4, or H.264) at its native resolution and progressive (if the camera actually records in progressive instead of interlaced). DVD video recorders had a brief lifespan in the 1990s when VHS was being phased out, but computer hard drives weren't yet large enough and CPUs not fast enough to do 1-2 hour video conversion in software. That's no longer true and even a lowly laptop should be able to convert the video in real-time.