One of the things that gives Jar-Jar his cartoonish appearence (his voice and actions have been proved to be something that can be worked around) are things like his Bugs Bunny teeth and his bright colours.
We know he can exist underwater with as much ease as on land so giving him traditional toady colours and losing the teeth seemed to make sense.
Amphibians can come in a huge variety of hues but giving him bright colours makes him look clownish before he even does anything which spoils the small amount of comedy he should be allowed to get away with (after his worst excesses are trimmed away).
One design I wouldn't want to change much but instead add more movement to is the Quarren.

He has the tentacles, the mandibles, an impressive brow, a squishy tongue and little cranial flanges.
However none of these seem to move much it would sell his general squidgy squidness more to put more motion in there.
His hands could do with a bit of a make over too (rubber gloves are good for washing up but they can limit the authentic organic nature of a beast).
The pale blue featureless eyes also look a bit fake, adding a pupil in there might improve an already nice design.