Having a tendency to fondle or caress; characterized by affectionate touching; cuddly or touchy-feely.
From 'fondle' plus the suffix '-some' (meaning 'inclined to' or 'characterized by'). The '-some' suffix appears in words like 'troublesome,' 'handsome,' and 'wholesome,' and is less productive today than it once was.
The '-some' suffix used to be extremely productive in English (making adjectives from verbs and nouns), but it's largely dead now—which is why 'fondlesome' sounds charmingly old-fashioned, a word that might have been common in the 1700s but feels quaint today.
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