When encrypting data that is not an even multiple of the block size, should I send the full last block?

This detail depends on the chaining mode you are using. The chaining mode determines how many times you call the AES primitive. and what, for a given input message. The simplest chaining mode is to simply split the input data into sequential 16-byte blocks and encrypt each one independently; this is called the ECB (as "Electronic Code Book"), and is known to have weaknesses (namely, if two input blocks are identical, which is often found in "real life" data, then the two corresponding output blocks will be equal friend, and everyone can see it).

Some chaining modes increase the data, meaning the encrypted message will be slightly larger than the input message. Other chaining modes (like CTR) do not. Almost all secure chaining modes require processing a "seed" which is a piece of data (usually the same size as the block) that should not be secret but should be known to both the sender and the receiver and should be different for each messages. Some modes (like CBC) require an even random IV, while some other modes will be happy with a simple counter. It is generally accepted to send an IV along with an encrypted message. You can also get the IV from the private key itself using a hash function.

These things are complex, and it is difficult to know if you did it right: security cannot be verified; a weak cryptosystem compiles and works just like any other application. It is not recommended to create your own cryptographic protocol. Generally. Using trusted primitives does not guarantee that the result will be safe.