Is it true that if I default on my second mortgage, but stay current on my 1st, the bank cannot foreclose on my house?

You must pay your second mortgage or risk foreclosure.  A second mortgage is secured by the property, the same as a first mortgage.  What you need to do here is to negotiate with both mortgage holders and have then agree just how they will split the proceeds of the short sale (first, however, they must agree to it).  Additionally, as part of such a sale, negotiate to have them waive their rights to any deficiency judgement.  That means, to the extent that the mortgage balances are not covered by the short sale, they cannot then look to you personally for the difference or "deficiency".