You made some great deals there. The Hawkins alone is worth over $300.00 in most cases, although I have picked two of them off auction sites for $150.00 & $200.00 so you really got a deal. The Thunderhawk, while older technology was a great inline. I remember when they came out, how people raved about their accuracy.
I was surprised that you said you were shooting Pyrodex pellet with a #11 cap system. If you hunt where it is damp or temperatures change a lot, you are going to experience problems shooting them pellets. I would not waste the money to convert the Thunderhawk to 209 ignition. Leave the rifle the way it was designed and shoot loose powder. The #11 cap is an excellent ignition source on loose powder, they are very weather proof, and loose powder is cheaper to shoot. Its a win win deal all around. Pellets, while convent, are over rated. You could actually try that JSG FFg in your Thunderhawk as well. Then you do not have to worry about swabbing. Give that a try. Any problems you are facing now I would suspect is because your shooting pellets with a #11 ignition. It is just not recommended because of the lack of consistancy.
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Get some loose Pyrodex RS and load that with 100 grains of RS and what ever the Thunderhawk likes. That rifle will then fire without flaw. Be sure to prepare the barrel properly before you load it for your hunt, and you will not have problems. I hunt with a Black Diamond XR. I have the option of all three ignition systems and choose to hunt with the #11 cap. I find with the loose powder, that #11 not only makes the rifle go off perfect, but there is little to no crud build up.
As for the T/C Bore Butter.. it is a good conical and patch lube in the Hawkins. Other then that I have little use for it. The T/C #13 cleaner is junk IMO. You could get the same results at a cheaper price with a bottle of window cleaner from the Dollar Store, maybe better results. Clean the barrel spotless like you would a center fire or shotgun and then protect the bore with a coat of good gun oil.Good bore cleaners is Birchwood Casey#77 or Butch's Black Powder Bore Shine. Try them, they work. Goodgunoil.. Birchwood Casey Sheath a.k.a. Barricade, Rem Oil with Teflon,andBreakFree CLP.Those oils will protect your rifle bore.
As for how many shots before cleaning.. On the range, swab the barrel (not clean the barrel mind you) between every shot. Remember, a damp patch with Windex on it, followed then two or even three dry ones. Then load just as normal. Also open that breech when you swab. As you work the patch, can you hear air moving through the nipple? If you can that mean that is also clear as well. Listen to your swabbing as well as look.
Before I hunt, I like to see how two shots work, without swabbing. I've seldom ever shot more then that. Once, I shot three times in one after noon. I filled three deer tags of course, but I never swabbed between any of the shots.