If Logan got cancer (almost anywhere), amputation would definitely be an option.
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Ah, but as we are seeing here, he would not regrow what had been amputated. His bones are not healing.
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His bones being adamantium replacements, regeneration would not be a problem. Huge amounts of cancerous tissue (and a surrounding buffer of unaffected cells) could be surgically excised and simply allowed to grow back around or within his skeleton.
X-Men Annual 11 answers this when Wolverine turns up late one night drunk, and says he can get drunk, but the alcohol is a poison that will be neutralised by the morning.
What I dug from a storytelling/character-building standpoint from way-back-when was the depiction of him drinking and smoking and seeing it simply as a reflection of his rambunctious character, never recognizing until later revealed that it was (at least partly) a consequence of, well, there being no consequences from them for him :)
X-Men Annual 11 answers this when Wolverine turns up late one night drunk, and says he can get drunk, but the alcohol is a poison that will be neutralised by the morning.