The problem you're likely facing is that you went from a SD TV to an HD TV. But the DVD players didn't change. So before your DVD player was outputting 720x480 to a TV with the same native resolution. Everything would look rather good. Now that same DVD player is having to output that signal to a TV whose native resolution is maybe something like 1280x720. So quite a bit more. The TV is doing the up-conversion, and well none are really that good (even the high end models). So it would be expected that you would see some quality loss.
The way around it is to get an up-converting DVD player, not just a new regular DVD player. Something with HDMI out. Blu-Ray players are all up-convertors as well as BluRay players. So if you went with one of those that would be fine. The up-convertor player will have extra hardware in it designed to help with the up-conversion so it will make it look better than just a TV would.
Any of the main brands are fine. Sony set-top models, Sony PS3, Panasonic, LG, Sharp, Toshiba, Samsung they all make sound BluRay devices. Just make sure it has an Ethernet port so it can be updated. And it should be profile 1.1 or higher. If you go with an up-converting (But not BluRay) DVD player, again stick with big name brands you trust.
It still might not 100% as good though. At the end of the day you're having to invent information that doesn't exist. So it will only get so good. Also, if your old TV was a tube (CRT) set, then that might also mean a degrade in quality now. CRT TVs are AMAZING in quality. But they're heavy, bulky, and not environmentally friendly. So they got ditched for LCDs, which are better in those regards, but actually don't produce as good of an image. But most consumers were willing to deal with that face in exchange for the small form factor.