I believe they're filter caps -- on the output side of a DC/DC converter, they help smooth any AC ripple that may exist.
As for parts sourcing, search for Aluminum Electrolytic SMD 680uf 10V (in Mouser or DigiKey, for example) -- should come up with lots to choose from for less than $1 each (plus shipping, of course.)
Have you measured for resistance directly across the caps (without voltage applied)? If they're shorted, they could be your culprit.
Or, when the board is energized (power applied to it), If you read voltage across the capacitors they're not likely the problem. Measure between TP1.2VS and ground, you'd have 1.2VDC -- that would mean C9G50 is likely good, or open. Between TP5VSL and ground, you'd have 5VDC and C9G52 is likely good, or open. And betweenTP3.3VS and ground, you'd have 3.3VDC with C9G51 being good or open. Always take care when working on live circuits.
Also take care when removing/replacing smd's. An smd rework station is nice to have, but if you're very careful soldering you can remove and replace them without damaging the circuit traces/board.
Re: the misshapen diode -- if it got hot enough to melt glass, the diode would've blown long before that. And if it didn't, the circuit traces would've let the magic smoke out.