Years ago I bought some "whiting Compound" from Brownells.
http://www.brownells.com/gunsmith-to...-prod1133.aspx
The same process that removes oil from gun stocks also works well for balsa. Its a powder you mix with a solvent (I've used acetone, alchohol, MEK, etc.) into a paste that will dissolve/thin the oil. It then draws oil out and into the powder. Some heating also aids the oil transfer. Leave alone until the powder darkens. Scrape off the oily paste and reapply until the the paste dries white and free of oil. Sometimes I'll use a flat iron to heat the dried paste to see if more oil will leech from the wood. When no more oil will come out you can glue, sand, or apply heat coverings to the balsa (I haven't tried paint). It also lightens the balsa nearly to the weight of when new.
I've used KR2, prior to the discovery of whiting compound. It worked as well in the old days. But, with recent formulation changes, not so much. It's also more expensive than whiting compound if you do a lot of repair work.
Cheers,