"After filling them, a few dissolved in places due to the oily-ness of the mixture I'm assuming."

Don't know anything about vegan caps.
So, I won't jump to assumption.

But the "oily-ness" is not to blame with gelatin caps.

They are water soluble.
The culprit is water.

Might have to reheat your oil and finish the boil.
Take it up to 110 C. and wait for the bubbling to stop.
Then, cool it to room temp and fill your caps.
(If you chill it, condensation water as it warms up will melt your caps.)

Then store them in a sealed jar with a desiccant.
Dry, uncooked rice is a suitable desiccant.

Aloha,
Weeze