I didn't know that valve cores were a thing, so I checked that out - but it turns out the tube didn't have a removable core. So I put in a new tube and all is good. It turns out there was a very small puncture about an inch from the valve (not where the valve joins the tube, as I originally thought). VegasTriker may have hit the nail on the head...I'm thinking maybe I might have pinched that area when I mounted the new tire, and for some odd reason it took a week before it became bad enough to do any serious leaking. Thanks to all for the advice!