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My ex who I share a child with recently started dating someone new roughly a month ago and it's going well, but also very fast. At first, I struggled with unresolved emotional attachment to her, but now that has subsided and turned to resentment. Trying to get her respond to texts and/or calls regarding meeting up to pick up my daughter has become more difficult as she is completely focused on the new man. I believe she is in the worshiping phase of this new relationship, which leaves me completely nowhere in her eyes. Is this familiar to anyone else?

This is not uncommon.  She's an adult so you have to let her live her life her way.  Since the marriage is Ended, so has your relationship, except as parents.  However, you do have a valid concern about parenting matters such as exchanges, etc.  Do you have a solid order where she cannot wriggle out of standard expectations in the parenting schedule?

If she fails to follow the steps for parenting exchanges, be prepared to get documentation of such failures.  For example, if she is a no-show or non-compliant, you may need to call the police and ask for assistance.  Have a copy of the court order with you.  The police won't enforce the order but as authority figures they may gets results you can't get from her.  Make sure they file a report and later get a copy as documentation in case you do end up back in court.  Police reports are much better than your own statements.

She's in a new relationship.  If and when it ends she may go back to her prior patterns of behavior.  Some parents, generally fathers and rarely mothers, seem more interested in adult relationships than parenting.  If that's how she is, then count your blessings, most disordered mothers seem to view their children as entitlements and weapons.

Is she ignoring your attempts to get HER to see the child, or ignoring your attempts for YOU to see the child?

If you are the one trying to see your kid and she just won't respond, then you are going to need an enforceable court order.  If you don't have an order yet, you can try texting statements and not requests - "I will be at your place at X time on Y date to pick up kiddo".  Then show up.  Document all the times that you tried to contact her and that you tried to see your child, and document what happened (she didn't respond; she wasn't at the house; she told me no, I couldn't see kiddo).

If you want her to come see the child and she isn't...there's nothing you can do about that.  We can't manage her relationship with her child for her.

Yeah, this is tough to transition into... The emotional stuff is especially hard on the beginning. 

What's the custody order, and how is she violating that order?