What a great question!
Nonprofits are always hard-pressed with resources - I understand your pain! It's rough when you're doing good work and desperately need the help of qualified marketing professionals...but barely have the funding to bring in a totally unqualified college student...or worse, an intern.
When it comes to looking at freelancers, you're right - it is difficult to decide which ones are qualified, much less trustworthy. And if you go to a site like eLance or Odesk, all too often the quality is low (as you've already experienced) or there is a language barrier that cannot be resolved.
However, your budget isn't quite as bad as it seems. As others have calculated, assuming a 40-hour workweek, you're looking at $400/week, $1600/month.
I'd recommend that you avoid hiring someone on a full-time basis, at least for now, and instead look at how you can make best use of that $1600. It might take a bit of time, but it would be well worth the investment to ensure that you have a solid marketing plan - ultimately if you have a strong marketing plan, that is what will help you with funding development, and may eventually yield enough funding to cover a more qualified employee.
For example, in the first month or two, you could hire an outside marketing consultant for a few hours to work with you to develop a plan. In another month, you might have the consultant train a staff member to implement some of the simpler strategies the consultant has recommended. In another month or two, you could have a new web site built (if that's needed). There are so many ways to wisely utilize the resources you have available that I think it would be better put to use with a good consultant who can guide you and help train your staff in the implementation strategies.
Some consultants may be willing to extend a discount on their regular rates for non-profits, especially if your cause speaks to them personally.
I would be more than happy to talk through some of these options with you and let you know how we can help, if it turns out to be a good fit on both sides.