Thrifty? Good, otherwise, I'd ask you to donate it to me, so I can repair it and send it to someone else!
To find the leak, the easiest way I found, is to rig up an air nozzle into a tee, and pressurise the core, as you dump it into water. You'll find the leak right away. A bicycle air pump is fine.
You can then plug the leak the best way you see fit: JB Weld applied on the leaky spot, and/or silicone the entire area.
Here's a pic of my contraption, in 1/2" and 3/4":
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