How can I deal with the anxiety of a new relationship?

First, let me note that without hearing all the details of what you are experiencing the following will be just an overview of possible helpful tools you can utilize.  I want to encourage you that new relationships are often plagued with many questions.  Most of the time those questions and ruminations come from insecurities we have about ourselves- to some extent we all have some level of insecurities that we must address. 

The most important thing that can alleviate some of those worries is open communication.  Talk with your partner about those internal worries you struggle with- if you can be open about some of those things perhaps the reassurance in the moment could help you better cope and move forward away from some of that rumination. 

It is also important to perhaps explore where those worries originate.  You can work on this by identifying the things you have a hard time with after you say them and work to think through why you think they will ruin the relationship.  It is important to think through this on your own (because you know more about yourself than your new partner) but also talk through the thoughts with your partner.  This allows that communication that is also so very important in a new relationship but also allows you two to get to know the ins and outs of one another in the process.   

It may also be beneficial to think about what relationships (either romantic or platonic) you have had that have shaped how you view yourself and your partner within the relationship.  This could be relationships with just about anyone you were close to growing up: a parent/guardian, sibling, teacher or other adult that had great influence over your life.  If there was trauma involved in one of those relationships that brings another aspect into the picture.  I would encourage you that if you are able to pinpoint a relationship that you feel changes/changed the perspective of how you view your current relationships to utilize this platform or another form of therapy to address those concerns and influences so that you are able to better communication and allievate those continuing thoughts.