(Bitcoin Core, recovering my wallet) problem finding the address I sent coins to in Bitcoin Core 0.11.2
First you need to backup your wallet.dat file. Then, since your installation is from a while ago, I would recommend reinstalling bitcoin, and then importing your wallet.dat.

Backup wallet.dat

To copy your wallet.dat on a Mac:

Cmd+Space then type terminal

In terminal type:

cp ~/Library/Application Support/Bitcoin/wallet.dat ~/Desktop

I would recommend also backing up a copy to another computer or disk drive.

Install Bitcoin-core

Download and open .dmg file from bitcoin.org, and follow installation instructions.

Restore wallet.dat

To copy your wallet.dat back on a Mac:

Cmd+Space then type terminal

In terminal type:

cp ~/Desktop/wallet.dat ~/Library/Application Support/Bitcoin/

Then you can export your key to whatever wallet you want.