This is one of those situations where you have to choose: will it be fish or fowl? Either is better than a mash of both together!
You want to keep the existing dryer unchanged, and extend the existing circuit ... stick with the same number of conductors.
You want to be compliant with the newer codes, make a new run with the additional wire, get a new cord for the dryer, and DISCONNECT THE BONDing strap inside the dryer.
Just using 'new' wire for the extension, and having a new receptacle that LOOKS like it has a neutral, but does not, is the worst possible way to do things.
