I want my python application to be able to tell when the socket on the other side has been dropped. Is there a method for this?
Short answer:

use a non-blocking recv(), or a blocking recv() / select() with a very
  short timeout.

Long answer:
The way to handle socket connections is to read or write as you need to, and be prepared to handle connection errors.
TCP distinguishes between 3 forms of "dropping" a connection: timeout, reset, close.
Of these, the timeout can not really be detected, TCP might only tell you the time has not expired yet. But even if it told you that, the time might still expire right after.
Also remember that using shutdown() either you or your peer (the other end of the connection) may close only the incoming byte stream, and keep the outgoing byte stream running, or close the outgoing stream and keep the incoming one running.
So strictly speaking, you want to check if the read stream is closed, or if the write stream is closed, or if both are closed.
Even if the connection was "dropped", you should still be able to read any data that is still in the network buffer. Only after the buffer is empty will you receive a disconnect from recv().
Checking if the connection was dropped is like asking "what will I receive after reading all data that is currently buffered ?" To find that out, you just have to read all data that is currently bufferred.
I can see how "reading all buffered data", to get to the end of it, might be a problem for some people, that still think of recv() as a blocking function. With a blocking recv(), "checking" for a read when the buffer is already empty will block, which defeats the purpose of "checking".
In my opinion any function that is documented to potentially block the entire process indefinitely is a design flaw, but I guess it is still there for historical reasons, from when using a socket just like a regular file descriptor was a cool idea.
What you can do is:

set the socket to non-blocking mode, but than you get a system-depended error to indicate the receive buffer is empty, or the send buffer is full
stick to blocking mode but set a very short socket timeout. This will allow you to "ping" or "check" the socket with recv(), pretty much what you want to do
use select() call or asyncore module with a very short timeout. Error reporting is still system-specific.

For the write part of the problem, keeping the read buffers empty pretty much covers it. You will discover a connection "dropped" after a non-blocking read attempt, and you may choose to stop sending anything after a read returns a closed channel.
I guess the only way to be sure your sent data has reached the other end (and is not still in the send buffer) is either:

receive a proper response on the same socket for the exact message that you sent. Basically you are using the higher level protocol to provide confirmation.
perform a successful shutdow() and close() on the socket

The python socket howto says send() will return 0 bytes written if channel is closed. You may use a non-blocking or a timeout socket.send() and if it returns 0 you can no longer send data on that socket. But if it returns non-zero, you have already sent something, good luck with that :)
Also here I have not considered OOB (out-of-band) socket data here as a means to approach your problem, but I think OOB was not what you meant.
How to tell if a connection is dead in python