I've posted about these a few times, and most of my "memorable" encounters were only memorable in terms of what a pain they were.
Questions that always come up:
1) Can a person with Freedom of Movement actually swim, or are they relegated to walking?
2) Why is it easier to swing a Spiked Chain or Ranseur, or even a Morning Star, than a longsword? (First three are Piercing and do normal damage, last is Slashing and does half.)
3) Successful Swim check means 1/4 movement, which is a pain for creatures with 30 feet. How do you handle that?
In your cartoon example, "Mr Stabby" would work just fine as a stabbing (i.e. Piercing) weapon. It's Mr. Slice and Dice that presents a problem.
Our solution to the combat situation was to change the rules slightly. It's "Thrusting" weapons that do full damage, a category based on how they're used rather than the type of damage they do. You can bludgeon someone with the end of a staff, when you jab with it, and that makes it a thrusting weapon. Similarly a Longsword can be used as a Thrusting weapon.
The caveat was, if it's not normally used as a Thrusting weapon, the character only gets half of their Strength bonus to damage.
The combat ritual should be:
1) Make a Swim check, to see if you can move or use Dex to defense this round.
2) Make Con checks, as needed, if the PC is holding their breath.
3) Make Attacks and or Moves as desired. Note that movement is 1/4 for a single move. For a 30 foot base move creature, that's one square one round and two the next. (PITA!)
4) Make Concentration checks for any spells being cast. (Note that none are needed if the caster can breath under water.) (Secondary note: First spell cast better be one to let caster breath under water.)
Also be aware that there are no 5 foot step maneuvers under water, unless the character has a swim speed or Freedom of Movement.
Role Playing scenes in an underwater setting aren't that different from RP anywhere else, so meeting with the Marlin King isn't any different from meeting with the owner of the Oakland A's.
Over all though, the best approach I've found to underwater adventuring is to ignore the fact that the party is under water, until and unless combat comes up. The fact that it takes them four times longer to cover a distance just means that more time elapses between events. Since we only really deal with the events, who cares? So long as they have a way to breath, let the game play out.