If the sets feel cheesy to you, you may have missed what the makers of this gem were trying to achieve. So much more than nonsdescript fakery, this sort of stage setting was at one time considered tres chic high style. --You may use 'Red Garters' as a reference here, or even Noguchi's designs for Martha Graham, if it helps validate the experience for you. This stylized vernacular particularly suits a filmed comic strip comedy/fantasy in which characters prance as if possessed and half the dialogue sounds Seuss-rhymed and sung. (Which is to say, anyone gleaning this film for indicators of "What is Reality and Truth?" has come to the wrong place anyway. --Even if the film's implied answer, which is something about Love, Faith and Hope, may be close to the mark.)
The film's designers showed excellent taste and instinct, in my opinion, in appropriating this style for Lil Abner. It seems not just right, but wholly inevitable. (See Altman's 'Popeye', by the way, to witness "real" locations sabotaging the suspension of disbelief that is absolutely necessary for a film of this kind to work.)
All in all a wonderful film, blessed with typically infectious Johnny Mercer songs, that rises above mundane political equivocation and essentially un-illuminating shoe-gazing.