No, there is no change to the waiver priority order. For the first month of the season, it's reverse standings order from the previous season, after that it's the current reverse standings order at time of waivers. (Successful claims have zero influence on waiver priority, that's only a fantasy sports thing).
Teams who successfully make a claim must put him on their NHL roster, unless the original team (in this case, the Sabres) are the ONLY team to submit a claim, in which case then the Sabres could send him to the AHL. But let's say the Sabres and another team that is lower on the priority list (let's say the Bruins) both put in claims, the Sabres would be awarded the player and could not send him down without clearing waivers again (there may be a limitation on when they can do this, I'm not 100% sure on that).