Should I leave my job due to my negligence?

I've been working as an yst for 5 years now in various companies but started my current position almost a year ago. Everything was going fine until a few months ago, it was discovered that I had made some errors in some company reports which then needed to be restated. This was something I truly worked on correcting but it seems that a few more errors popped up in the following months. This last week, it turns out that I made another error that is causing my coworkers to redo some work as well relating to a budget change. I am not in any way purposely doing anything wrong, I just seem to not do things in the way the company expects and its hurting those around me. At this point, my boss is delegating projects that may have gone to me in the past to a co worker, I'm uming based on his thoughts that he cant trust my work. My dilemma lies in the fact that I'm now looking for a job, but I would rather apply for other positions while working here so that they cant contact this company for a reference. If I leave now, I will not be able to state that I don't want this new company to contact my former employers as I don't work for them anymore. This has never happened at any other positions I've ever had, nothing but glowing recommendations from former supervisors. Any thoughts would be appreciated.