How can I overcome feelings of unworthiness after finding mistakes in my research?

I'm going to suggest you attack this on three fronts:

1) As you progress as a researcher, I think you will come to realize that minor errors in published research are actually quite common. I define minor errors as those that don't affect the conclusions of the paper. See this question/answers: How common is it for submitted papers to have minor errors?

2) Understand that corruption is actually about intent and secrecy. High profile cases of corruption and ruined careers only generally happen when the researcher or the study is famous and the problem involves purposefully lying. You can easily avoid this type of situation by putting a corrected version of the paper on your website. See this question/answers: How does an Erratum on one's own paper affect one's career as a researcher?

3) Get a handle on how you think about problems like this, which are relatively common in the life you have chosen. I would suggest that there is a long way between "reasonably cautious"/"feeling embarrassed" and "extreme anxiety"/"failed my life goals and my moral values"/"bad person that does not belong in academia." I am going to strongly suggest seeking out your university counseling services, if available. See my answer to this (only somewhat related) question: How do I know that I have truly lost interest in research and should drop out of a top CS PhD program? Or similar: How to deal with fear of failure as a PhD student?

Sure, we all want to publish error-free papers, but mistakes happen sometime, and when mistakes happen, the best you can do is be proactive to get the correct information out there (and do better next time so you don't make the same mistakes again).

I know of one particular case in my area of research that sort of matches what you are experiencing. In 1997, Derek Shaeffer, who, at the time, was a PhD student at Stanford, published a paper with his advisor, Tom Lee, in the IEEE Journal of Solid-State Circuits. Then, due to some errors in the original paper, not one, but two correction papers were published to correct the mistakes of the original paper.

Things apparently turned out OK for Derek: he is now the Director of Engineering at Apple in Silicon Valley.