Hi,
I did a test on an Alesis Trigger IO and I have the same problem.
I think that there is a kind of signal sent to the ring through the ground. Maybe I shoud add diodes on the ground line to avoid this?
Don't know what to tell you..other than If you are sure the Foam Columns are making contact with the Head.. start eliminating pieces in the entire circuit...it changed when you replaced Piezos unless you replaced at the Jack and made a mistake somewhere...the Sensitivity Board is pretty straight forward and marked.. + or -
To trouble shoot...Bypass the Sensitivity assembly..
It's basically an Audio Taper A100K Stereo rotary Pot..Horizontal 7-pin with resistors.
The Sensitivity PCB assembly output is wired to the Tip/Sleeve on the PCB on the Jack.
The Rim Piezo (Purple/Blue) is wired directly to the Ring/Sleeve on the PCB on the the Jack.
..De-solder the Red/Black wires (Center/Head Piezo) and tie them together....you can also check the Outer Piezo (Blue/Yellow) the same way..
If the snare works as it should then it's the Sensitivity assembly/PCB Board
If the issue remains it's probably the Jack which also has a Circuit Board , resistors and a lot of Glue covering everything.
There some additional SMD resistors in place to cool down the signals.
It's not uncommon for something to break internally or get dust/debri in it.
I don't know of anything else strange a Piezo can do except sometimes you'll find one with Reverse Polarity.
Piezo's are pretty simple they either work or they don't (Broken wire..Broken Ceramic) .....some may put out less Voltage.
EDIT:Added Image.