Replacing the shower head will not stop a drip, Repairing the diverter or replacing the diverter spindle and seat should stop the drip.

Depending on the make and model what kind of replacement parts are needed

Is this a shower or shower/tub?

Does the it drip when "off" or do you mean that when using the shower that while most of the water comes out as a spray some drips off the shower head?

The shower head drips water when off. Please explain the diverter, etc. Wouldn't it be easier if I just bought a new shower head?

A divert does just that it diverts the tub (bath tub spout) water to the shower head if this is a used for both.

The shower head is not used to shut off water it is a spray only using the water coming from the faucet 2 or 3 or 4 valve body

The facuet is leaking. Just like a dripping faucet on a sink.

The only difference is that instead of being connected to a spout it runs up the wall to the shower head.

You need to replace the washer/seals/cartridge on the faucet.

Give some indication of the age and type (separate hot/cold, single lever that controls both temp and volumne, single control that only controls volume, 2 controls - 1 temp and 1 -volumne) and the brand if it is marked.

This a single handle control in a stand alone shower (no tub). I have already replaced the washer that goes from the neck to the shower head but when the shower is off I still get a drip.

Get the name of the shower valve and then replace the inner parts and that will stop the drip.

YOUR water bill must be sky rocketing

Do you mean the handle part? or the shower head itself?

The part behind the handle that controls temperature and volume is called a shower body.

There should be a name some place on the cover or handle.

If no name is located try to take a picture and take it to a local plumbing supply or hopefully a big box store that has repplacement parts