I am not burned out. As long as any Souls game has the same quality level of Bloodborne or Dark Souls 3 or better, then I would gladly play more souls games. I actually want to continue playing Dark Souls 3 but I've exhausted all of the content, so I am waiting until the DLC.
It's understandable if someone is burnt out, though. There have been 5 games. I usually stop playing a Souls game part-way through for one reason or another, and then pick it up again later, but this time I blasted through Dark Souls 3 in a couple of weeks.
Sometimes I get series fatigue with games in general; for example, I started to get burned out with Final Fantasy later in the PSX era. I couldn't really continue playing through any more of them with great interest until a few years ago. A break was needed, and I was able to come back and enjoy some of the games I didn't enjoy in the past.
Also, Miyazaki feels the same way, he is "done" with Dark Souls for now. So after the DLC it will be something new.
DS3 does everything well, but feels creatively exhausted.
Its not the series, it's the game itself that's lacking. A key part of the appeal of the games is the sense that you are exploring an unknown and mysterious world, DS3 feels more like a nostalgic theme-park. Its not too damaging the first time through because at least you have bosses to anticipate, but on replay when you know the lay of the land and where big encounters happen it falls flat.
At which point I found myself keenly aware of how lacking in ideas, character, and innovation this new entry was... and my interest fell off a cliff.
What's doubly damning to me is the feeling that there's very little FROM can do to make DS3 a title that stands on its own feet with patches and DLC. Its simply too mired in past glories to ever become its own thing; its constant call-backs to past glories simply remind me of how great and memorable those games were, whereas DS3 itself is fairly anonymous and uninteresting.
I guess it's understandable if people feel this way. However, I strongly disagree. I enjoyed this game immensely, and I don't think that having a strong connection with DS1 is a problem. If you really only enjoy 'new' worlds, then that might have that effect, but it doesn't for me. I generally like the game, I feel like it is not 'creatively bankrupt' simply because it is using some content with past games. ENB's reactions are kind of a good example to use. There are some fan-servicey elements to the game, however, they have built a new world and new lore with it. DS3 kind of 'completes the picture' when talking about the series lore in general.
I think your problem is more of a personal one, I would say, and not necessarily reflecting of every player.