That does sound stressful. I am the same way in that I was always taught that a job worth doing is worth doing well. Many coworkers disagreed and pointed out that we don't get paid enough to do THAT good a job. I would just continue doing as you have been, no more and no less. They can't expect you to do more when you obviously need to look for another job. That's on them, not you.
I worked for a company and we were negotiating to get Veterans Day as a paid holiday. The company refused and said that since we were paid by the hour, and not paid during our off time, we could pick any day we like and call it Veterans Day. It didn't need to be one specific day. So we took them at their word and everyone took off work every Friday to celebrate Veterans Day. After the fifth week of this, they decided it would be more cost-effective to limit our holidays to one specific date each year, and gave us Veterans Day as a paid holiday.
How many accounts will you and your coworkers be able to service if you need time off to go find a job, attend interviews, or even just be too sick to work? If they need extra help, maybe they should hire some temps?
One of our biggest fears is the fear of the unknown. Don't let it get to you. Things will work out, they always have and they always will.