Second of all, my sympathies for your plight. Working in an environment that you describe sounds like pure Hell. Unfortunately, sometimes there is simply nothing to be done about abusive work environments besides moving on.
You don't say what country you're in, so I can't comment on local labor laws that may protect you against such abusive behavior.
I'm throwing out my resume left and right but - obviously - my skills and desire to learn are being frowned upon.
Why are your skills and desire to work "obviously" being frowned upon? Sending resumes in a shotgun approach may get you another job, but it may not be much better than the curent one. Chances are that you may be coming across as desperate for another job, which is a big red flag to potential employers.
...during which time they will expect me to run a great deal of their business.
In your resume/cover letter and when you interview, focus on this- employers (should) welcome the responsibility that you took on while the bosses were away.