How much would it cost to ship a drawing?

The rule is to pack to protect.

You didn't say, but I assume your working on paper and I assume headshot badges are somewhat small.

The first thing you do is get an envelope about 1.5 to 2 inches larger all around than your art.

Get some light, inexpensive rice paper or use a soft, light weight paper -- the important thing is soft and acid free (hopefully you are doing your art in archival mediums). Place your art on the rice paper and carefully fold/wrap the rice paper around your art. Take care not to fold the edges of your art! If you have to use two pieces of rice paper front and back to cover all exposed sides, do so. You want this packet so your art does not move about.

Now, take two pieces of card board the same size that will fit into your shipping envelope. You tape your wrapped art in the middle of one side of the card board and sandwich the other card board over that. Tape the sandwich together. You don't need a lot of tape, just enough so things don't shift around.

Address your envelope BEFORE you place the protected art inside. Put in your invoice, seal it up, and you're off to the post office. If you are charging for shipping, be sure to include the price of the packaging, tracking, and insurance.

If you start to ship larger items, like canvases or framed pieces, you're talking some serious crating.