Me too!
I get sore palms every weekend after my usual 30 mile ride. On my last tour (10 days) my left hand went partially numb and the feeling didn't return for a few weeks afterward. Like you, I'm looking for answers, because this is the biggest factor that is detracting from my enjoyment of touring, and I LOVE touring. I've used "GRAB ONs" for probably 30 years. They're the black foam tubes that you put on your bars, above the brake hoods, as well as below on the drops. They help somewhat. The foam usually loses it's cushioning ability after a year or so, and I've replaced them dozens of times. It's a pain, because you have to remove the shifters (I have bar-ends) and the brake lever assemblies to slip on the new foam.
I haven't used pipe insulation. It's probably the same principal, and if it's thicker, it would be worth a try. It would also be good if you didn't have to remove the hardware to install.
On my last trip to the bike shop I saw what was, to me, a new product - Aztec Padded Bar Wrap and Tape. It consists of some "ergonomic" pads - thicker in some places than others - designed to put maximum padding where you need it most - and some foam bar wrap to wrap around the pads after they're installed. I bought it and tried it out. It was better than the Grab Ons because you didn't have to take off the hardware. The padding feels a little funky, but on my first ride it did seem to be an improvement. However, I still had some degree of pain on my left palm - the kind that would lead to numbness on a tour.
I recently saw a posting on Crazy Guy On A Bike by a guy who was doing the Adventure Cycling 30-Year-Anniversary Tour, and he bought his "dream bike" for the occasion. It was a custom made Jamis Aurora. One of the things that struck me was the handlbar he had installed. It's a FSA K-WING CARBON ROAD HANDLEBAR. It looked like it had big, semi-flat places to rest palms on the upper corners of the bars - right where I like to rest my palms, but where I can feel the round bars irritating my hands.
I thought this looked like a very intriguing solution. Except they cost about $229. That's an expensive experiment. I did a post asking if anyone had any experience with these things, but no-one wrote back.
I've considered trying to build my own little palm rests to attach to the bars, but can't figure out a good way. It seems as if building the bar with carbon fiber affords the opportunity to shape it any way you want - including the way the provides the most protection against the numb-palm syndrome we're experiencing.
You can now buy grips for mountain bikes that have a thick, oblong spot built into the outer section - it looks like what I've envisioned as a way to cushion my palms. It seems like someone could invent something similar for us drop bar users.