Unless you have (recent) European ancestry, you won't be able to get an EU passport, which will make it as good as impossible for you to find legal work in Europe. A work visa has to be organised through an employer, the catch being that an employer cannot just hire a non-EU citizen, they have to prove that they couldn't find a suitable EU citizen. Which is a fairly unlikely proposition. Things have been made harder still by the recession.
Note that if you come over on a Schengen visa you have 90 days in which to convert that (through an employer) to a valid work visa. If you fail to do so you have to leave not only Spain but the rest of the Schengen zone (which covers most of Europe) for a further 90 days.
So, what can you do? Your best bet is to get in contact with the American schools - either private language schools teaching EFL with a focus on American English or International Schools teaching the American elementary/high school curriculum. In the case of the former, the employer might be able to organise a legitimate work visa by stating the case for an American English speaker; likewise in the case of the latter but with the case further strengthened by the requirement for someone who was familiar with the American schools curriculum.
You definitely have jump through hoops to work legally in Europe; remember that Europeans have the same issue when trying to work legally in America.