Yes there will be huge improvements sonically when using a $400 DAC/preamp vs a $6k preamp/DAC. Your upstream electronics play important role as they are the ones feeding the signals to your power amp and speakers. If you feed them with garbage the overall end results will be garbage too. Garbage in garbage out.
Since you have a Classe CT-2300 power amp I recommend that you get the matching Classe CP 800 stereo preamp/DAC and the two will pair well together. Besides the CP 800 stereo preamp/DAC has been discontinued last year and nowaday it can be had for around $3500 to $4k used. This will be the cheapest solution for you. You can always get better ones but will probably cost more.
Your Classe CT-2300, which is basically the same amp as the CA-2300, is a wonderful sounding amplifier. You can also get a good quality tube preamp plus a standalone DAC/streamer to potentially change the tonal character a little bit.
Auduo Research, Primaluna, Cary Audio, Rogue Audio make great sounding analog tube preamps. I would look for a used one. The Rogue Audio P7 is highly regarded by many and I would strongly suggest that you look into that. But you will still need a standalone DAC with these tube analog preamps. Your Teac UD 301 DAC wouldn't cut it.
After you upgrade your preamp/DAC or preamp + DAC you might not even need to upgrade your speakers after all. Or you can always move the Vandy line if you like their sound profile. They are great sounding speakers IMO.