Maybe we're using the term hedge differently, but if I'm effectively short by selling a call contract, isn't my hedge a long position in the underlying? The reason I say this is because if I also short the underlying, then I'm on the hook for both the call delivery and short covering if the underlying goes up. I think this is right, but correct me if it's otherwise.
You are correct, I did mean you buy the underlying.
The way I see it is that if the stock has gone against me and my option writing position, what I should do is to use the premium proceeds to fund transactions shorting the stock. ...
You do not want to short for the reasons you stated above.
...If I short enough, I drive the price down.
That's assuming you have the ability to influence the market in a meaningful way and maintain that influence over time.
If I'm long in a downtrend, I need my hedging strategy to move prices up...
Here we may have a terminology issue. I understand a "hedge" to be an offsetting position to minimize losse when the price action moves against you. Not an attempt to influence the price action.
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