"Bilbo was enjoying his mead, ambling around the yard, not
accomplishing anything in particular this summer afternoon."
The origin of "amble" is from a Latin word meaing "walk about", so it would primarily denote a walk. It doesn't depict any parallel activity undertaken while walking.
As to the style and intent of the walk: to walk about suggests that the walk is not undertaken specifically to get from one place to another: "about" in this context feels like walking "to and fro", or perhaps "in a random way".
OED British colloquial. intransitive. To move or travel in a leisurely manner; [...] Frequently with around, along, about.
1988 Bicycle Midsummer 57/1 Pootling through Nottingham recently, I came across a branch of a shop called Concept Man.
2020 http://more-to.org/about/And I love walking, enjoying several expeditions in the greater ranges of the Himalayas and the Andes, as well passionate ramblings around the mountains of Wales. When not walking, you’ll often find me pootling about in the garden, or sitting out with friends and a nice bottle of wine…
To amble without real aim, yet in a friendly and harmless manner. It's not required to acheive nothing, though it is a frequent side effect.
Bimbling can be made a little more business like with a slight hunch of the shoulders. "Tron and Enid whiled away many a Sunday afternoon on a pleasant bimble round the shops."
Bimble (noun), (plural bimbles), (chiefly Britain): A gentle, meandering walk with no particular haste or purpose.
Bimble (verb), (third-person singular simple present bimbles, present participle bimbling, simple past and past participle bimbled), (chiefly Britain, intransitive): To walk with no particular haste or purpose.
"Bilbo was enjoying his mead, bimbling around the yard, not accomplishing anything in particular this summer afternoon."
While you may have a little difficulty transmogrifying the words, this is precisely the state of mind being conveyed by "Feelin' groovy". Even if this does not end up being what you use (which is likely :-) ) it's worth listening to the song and reading the lyrics to "soak up the ambience"
Video - Simon & Garfunkel - Concert in Central Park - Feellin' Groovy / '59th Bridge Street Song'
Slow down, you move too fast
You got to make the morning last
Just kicking down the cobblestones
Looking for fun and feeling groovy
Ba da-da da-da da-da, feeling groovy
Hello lamppost, what'cha knowing
I've come to watch your flowers growin'
Ain't you got no rhymes for me?
Doo-ait-n-doo-doo, feeling groovy
Ba da-da da-da da-da, feeling groovy
I got no deeds to do, no promises to keep
I'm dappled and drowsy and ready to sleep
Let the morningtime drop all its petals on me
Life, I love you, all is groovy
In Middle French, the noun "gambade" referred to the frisky spring of a jumping horse. In the early 1500s, the English word gambol romped into print as both a verb and a noun. (The noun means "a skipping or leaping about in play.") The English word is not restricted to horses, but rather can be used of any frolicsome creature. It is a word that suggests levity and spontaneity, and it tends to be used especially of the lively activity of children or animals engaged in active play.
'to walk in a leisurely or aimless manner; to amble, wander.' (Frequently with along, off, on, over).
Interestingly they note that the original meaning was 'to go away quickly or promptly; to make haste' but that this meaning is now rare.
So, perhaps the different usage depends on whether you use British English or American English?
As a British English speaker if, for instance, I was on holidays and I visited a nice tourist spot with my family I might say something like 'let's stop for a coffee and then go for a mosey' and in my postcard home I might say 'we enjoyed moseying around this tourist spot'. It would be understood that I had had a little wander around the town.