I’m a novice at audio…that being said I’ve made some discoveries. Don’t change everything at once. From information gleaned on the internet and on this site I am guessing improvements would come mostly (60%) from speakers, then speaker placement and room treatment (15-30%) then amp, DAC,
Since you and I listen to mostly jazz and Blues for me, I will say the Klipsch RP 500m does really well with the genre we like. I have the RP 150m which were the previous 5 inch from Klipsch. They do really well with sax, horns, cymbles and vocals and standup base.
If you listen mostly at your desk the RP400m are really quite good paired with a sub. My brother-in-law has them with a 10” Klipsch sub and that combo sounds really good. His listening seating is about 7-8 ft in a room a little bigger than yours.
My son-in-law has Elac Debut reference DBR 62 and an SVS SB1000, that combo also sounds pretty good.
Change your speakers and then listen for a few months, play with the positioning and them go from there.

Definitely amp and speakers with that budget. Speakers first. There are a ton of bookshelf options. I would look Wharfedale, Dali, Q Acoustics, and above mentioned Elac and Klipsch. Then find the right amp to power whatever speakers you choose. Just remember if you happen to buy speakers rated at 4 ohms make sure the amp is rated for driving such speakers. The Denon will get the job done in the meantime, but if vinyl playback is your thing, I would opt for an integrated that isn't loaded down with all the electronics meant for video. The phono stage on AVRs are usually an afterthought and not the best quality. Find a quality amp with a nice phono stage. I can vouch for Audiolab 6000a. Solid as a rock and beautifully balanced playback. But I'm sure there are others here who have great advice on other options. Good luck. Happy spinning.