I haven't heard your integrated or speakers, but unless they are horrible, I would upgrade your turntable. Most turntables come with a decent cartridge, so you can kill two birds with one stone.
Right at your price point, I'd take a hard look at the Marantz TT 15S1. It's very similar to the Clearaudio Emotion, which I own. It comes with the Clearaudio Virtuoso cartridge (which I also own, the wood version) which is a really nice MM cart.
Another option would be a used Music Hall MMF 5.X or 7.X turntable and a Hana SL cartridge if your integrated will handle MC carts. If you're happy with the speakers and integrated, that would be a big upgrade.
Looking at it another way, speakers will change the sound of your system more than any other component. Making an assumption based on your current speakers that you don't have a large space and you like monitors, you might consider a used pair of Harbeth P3ESR speakers. You should be able to find them used at your price point.
Right at your price point, I'd take a hard look at the Marantz TT 15S1. It's very similar to the Clearaudio Emotion, which I own. It comes with the Clearaudio Virtuoso cartridge (which I also own, the wood version) which is a really nice MM cart.
Another option would be a used Music Hall MMF 5.X or 7.X turntable and a Hana SL cartridge if your integrated will handle MC carts. If you're happy with the speakers and integrated, that would be a big upgrade.
Looking at it another way, speakers will change the sound of your system more than any other component. Making an assumption based on your current speakers that you don't have a large space and you like monitors, you might consider a used pair of Harbeth P3ESR speakers. You should be able to find them used at your price point.