Obviously the existing gaskets have been reinstalled at least once before, as a result I would replace them.
If you have not sourced these gaskets yet from your friendly Lehman Dealer, it may be that you can get the gaskets and the gum washer (if you need it) from Sen-Dure. I would not be surprised that Sen-Dure supplied the original cooler, they did on my Cats. Lehman, like Cat, likely does not make coolers.
Have a look near the bottom of Page 5 on this:
https://sen-dureproducts.com/Sen-DureCatalog.pdf
In order to clean out the tubes with the end caps off, (always a good idea so you can see whats going on) if the tubes themselves have an internal diameter of 3/16", 0.188" I would acquire a couple of 0.170" brushes like these:
https://www.birchwoodcasey.com/products/bronze-bore-brush.html
At the same timeI would get enough extensions to reach through the tubes and a couple of bronze brushes at 0.220" just in case (an additional $4 investment) that I could chuck in the lathe and grind down to precisely fit, if I did not like the fit of the smaller brush.. Perhaps it's better to buy a kit, but they typically have brushes that are too big.
To clean the tubes, assemble the extensions & brush, then chuck it in a slow variable speed drill motor. You could do it by hand as long as you can spin the brush, but it will get tiresome.
I would orientate the cooler with the cleaner end high and put a bucket under the clogged end to capture the junk.
With a trigger bottle of water, wet one tube and insert the brush, then start it slowly spinning. Run the turning brush all the way to the low end of the tube (but not out) while you spray water down the tube. Do not force anything, your drive is made of brass and will fail if forced. Let the brush do the work.
Now, how do they say it? Rinse and repeat!