What is the best approach when contacting a designer to avoid a negative answer?
If you can't afford a good designer, you can't afford a good designer.
Most designers I know of have a chip on their shoulder about being asked to work for too cheap, or worse, free. It shows a great disrespect for them as a person and for the work that they do.
A common mantra is "Work for Full Price or Free, Never Cheap." Working for full-price should be obvious. Working for free has its benefits, and many designers will work for free if the project seems worthy or at least really good for the designer's professional growth. Working for cheap devalues the profession and undermines the process, the results, and other professionals.
If you are somehow charitable or working for a good cause, maybe you can woo a designer or firm to work for free. If not, pay full price. The only way to get cheap is by luck; you won't likely woo a designer (or doctor, or lawyer, or any other professional) into accepting less money than they are worth. You may get lucky and find someone who doesn't know better though. ...or a designer who's just not worth very much (but then, their work probably isn't either).
I can say that I am not at all intrigued by a website that has taken ten months to produce no results, no investors, a low budget, and an unfinished product that I'm not allowed to know about. At the very least you have to tell people what you do, or intend to do.
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