The Studio Ghibli movie, Castle in the Sky fits that bill pretty well. Especially on the "old adventure vibe" front. It's definitely got cozy and funny moments throughout, but all the guns and explosions might be a bit much.

But if you need stuff that's pure cozy, Non Non Biyori is good, as is Yuru Camp. Not big on the fantasy or adventure part, unless you consider walking to the candy store, or going camping, to be adventures.

For something cozy and fantasy based: Hakumei to Mikochi is good too! It's about a couple of lesbians who live in a tree, in a charming fantasy setting. It's got big cottage core vibes.