Aloe root rot
I bought an aloe plant from a local grocery store and immediately repotted it in a larger container as it was over growing the pot it was in. I expected some shock from being replanted but it never fully recovered. The leaves look somewhat better but they have a strong foul smell when broken off that I’ve never experienced before. I think it might have root rot. Does that sound correct and, if so, is there anything I can do to save the plant?
It's challenging to diagnose a problem without several images. But when you wrote the plant/leaves “have a strong foul smell,” then, yes, it’s rot. Most common is a bacterial rot.
I suspect a series of events. For one, if you brought it home during one of the recent wet, cloudy, cool-to-cold days, it was chilled beyond its tolerance. Temperatures below 55 degrees F cause “chilling injury” on succulents. Might as well have been frozen.
In any event, the only way to stop it is to cut off the rotted part, allow the healthy part to air dry for a day or so, then set the good part onto barely moist potting mix designed specifically for succulent plants. You’ll need a good dose of patience while it roots in.
“Getting cacti [and succulents] through the winter is less a matter of tender loving care than intelligent neglect.” See the details at https://lancaster.unl.edu/hort/articles/2008/cactiwinter.shtml.