These galvanically isolated 1:1 baluns are hard to find on the web.
..but they are easy to make with a small ferrite toroid and non overlapped windings. Also, old Ethernet cards are a cheap source of high bandwidth signal transformers which are galvanically isolated. In this case its just needed to balance the input signals into the pancake coils to minimize the crosstalk.
You are welcome to try balancing the sides and see how much that helps. We are here to learn as much as invent, after all. So should we really need to be able to completely avoid any signals on the output without any water in the tube?
Not completely but as much as possible. If the S/N ratio of the NMR signal will be too small to observe over the crosstalk, then there is always TDM to the rescue. I cannot imagine that being possible with coils that close to each other.
They should not touch - that's for sure. Right now you don't even know whether the crosstalk is caused predominantly by inductive or capacitive coupling (or acoustic). If the coupling is predominantly capacitive, a grounded shield between coils, (made out of Litz cloth so it does not impede AC magnetic fields) will make a big difference.
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