My title will confuse some of you, because NRE (New Relationship Energy) can be very polarizing - some folks love being carried along on a tide of feel-good-brain-chemicals and excitement into the midst of a new relationship, and some people feel very out of control and anxious through that phase of relationships. This post is for those of you who, like me, feel the latter.
I've mentioned before that my biggest strategy to avoid this is to go slow in relationships -- it can feel like mild torture sometimes to space out time with someone you're really eager to dive in with, but it tends to lend itself to taking time to talk and think between meetings, rather than being carried on "this seems good" through evening dates that turn into weekends because you're SO INTO this person. In polyamory, you can do this relatively organically. Make plans with existing partners before pulling time for someone new out of the calendar; put your phone down (at least for the most part) during focused time with other partners, and you'll find that you pace yourself much slower than "oh my gosh they're so exciting let me talk to them whenever I'm not at work." (I can't be the only one with a bad habit of neglecting real life for flirtation if I let myself, right?) It has the fringe benefit of making sure you don't neglect existing partners in favor of the new shiny, as well.
Another thing you can do is take a step back for realism - make a list of what you actually know as a fact (not what you're extrapolating or imagining) about the new person. It can be helpful to recognize how much of those early intense thoughts are actually fantasy of how we hope the relationship will pan out - realizing that much of what you know is actually publicly available information like name, employer, address (or that you don't even know that, just their hair and eye color and height and that they like whatever the thing you have in common is) can be a good way to slow your fantasizing down and remind you to ask compatibility questions. On the other hand, if you're Extremely Polyamorous and open with a slew of compatibility questions about relationships, realizing you know what their idealized polycule structure is but not what music or tv they like or the sorts of vacations that sound fun to them can also be a reality check. Both things are important and taking a minute to say "okay it's only been three dates and I don't know that much, let's not make up cute couple names yet" can make it easier to calm anxieties about where a relationship might go or not. So can just recognizing where the gaps in your knowledge are and reframing the getting to know them process to include that better.
I wish I could say that examining our mononormative dating biases made NRE and expectations of escalating relationships easier (and for some people it seems to!) but for me it mostly gives me a new layer of worries about whether my expectations for relationships are still reasonable and whether they'll match the ones this person has decided are important as they do the same. Or, alternately, whether they'll be disappointed at how few of those expectations I still think are necessary, or have an issue with how I frame that. The "I know they aren't magic or perfect but I still WANT them to be, because of how good it feels to spend time with them," is a big anxious-limerence-NRE-mood for me, and the majority of why I don't like it. If you relate to this, considering which parts of these worries are not about you (are just structural) can be kind of a relief, and time passing where you get to know the person and understand that you can actually communicate well together can help. Having your first disagreement and resolving it well (showing you can have hard conversations in a functional, if not compatible, way) can help form the background security to let you keep enjoying the early phase of a relationship without feeling dragged into a current by NRE.
Overall, all of these suggestions boil down to remaining grounded in your life, finding sources of internal and external security to lean on, and giving yourself grace and time when you're in NRE can make it less overwhelming. What do you find helps you cope with moments where the intensity of a burgeoning relationship feels like a current carrying you away?