Sounds like you have a particularly picky player. Are these Verbatim +R disks by any chance? You can bitset a +R dvd to make it look like a commercially made DVDRom disk, and thus increase compatibility. Not all dvd burners can do this, so we’d have to know which one you have and what program you are using to burn the disks.
Verbatim is recommended because it seems to be the most reliable, most consistent media available right now. There are others that are well liked, but aren’t as easily found, especially in the local shops.
DVDShrink is obsolete as a dvd ripping program. There are many dvds that you cannot decrypt and rip to the hard drive using Shrink. I suggest getting a free decryption program that is still updated: DVDFab HD Decrypter. http://www.dvdfab.com/free.htm
DVDShrink is still a good transcoder for compressing your movies to fit onto a single layer dvd, so there is nothing wrong with continuing to use it for this purpose. There are other options, including some that will produce better output. Any good encoder will beat a transcoder like Shrink, but depending on the amount of compression necessary and the size of the screen you are watching on, you may not notice the difference. I tend to use an encoder just so I get the best possible output on my copies.
If you are interested, look up DVDRebuilder. There is a free version and it comes with a very good quality encoder called HC. The main drawback is that any encoder will take more time than Shrink. A modern, dual core machine will cut this time down significantly…it takes roughly 80 minutes to run Rebuilder on my main machine. I just copy the main movie though…no menus or extras. This cuts down on needed compression.