I have recently purchased a new Bialetti moka pot and I've found that after brewing a cup of coffee and letting the pot cool down, there's coffee in the bottom chamber. Am I cutting off the heat too early? Or is this some kind of issue with the pot I'm using? Let me know what you think.
I think you are on the right track with your idea. To understand what's happening you need to understand how the moka pot works and how to correctly use it. I'll explain it here quickly but there are some nice illustrations on how it works on the Wikipedia entry for it.
First, the heat should be cut off once you hear the first hissing sound. That is when all the water has been pushed through the ground coffee and the brew is ready. By construction there remains a little bit of water in the bottom chamber. That is because the spout of the filter basket doesn't go all the way to the bottom of the chamber.
When the water starts boiling, the steam of the evaporated water pressurizes the chamber between 1,5 and max. 3 bar (that is the pressure the safety valve sustains without opening) depending on the grind size of your coffee, the amount of coffee and if you tamp it (which you should not). The gas pressurizes the area above the water, which pushes the water through the spout, through the ground coffee and into the top part until the water level is so low that the steam exits through the spout. That's the hissing sound. So one of two things could be happening here:
There might still be a lot of water in the "brewing chamber", where the ground coffee is, when you cut off the heat. Then the pressure in the bottom chamber will fall and the coffee in the middle gets sucked down into the water chamber.
Pretty similar but instead of the ready brewed coffee it is some of the grounds that get sucked into the water chamber and keep brewing there in the rest of the water. It may be that it is ground too finely.
Both of these things are not a problem though, just flush everything with warm water and that's it (no detergent if it's an aluminium pot or you risk oxidation). It only affects the coffee if you cut the heat way too early or too late.
Probably yes, prematurely cut down heat may be the reason.
During normal operation of a moka pot, all of the water filled in the bottom chamber evaporates and flows through the top chamber through the coffee grounds. If the level of the filled water is not above the marked safety level you can safely wait to be sure till all of the water evaporates. You can understand this easily as bubbling sounds end and you can only hear a stable hissing sound.
When heat is cut off prematurely some vapor may form droplets in the middle chamber and may flow back to the bottom chamber. However, it should be lightly brewed and the amount should be really small. If this is not the case, your pot may have a leak somewhere.
One last option, which is not clear from your question is, whether you mention coffee solution in the bottom chamber or ground coffee particles. If you observe ground coffee particles either in the top or bottom chamber, it indicates that the grounds are too fine for a moka pot. You should ask your roastery to grind them coarser.