Why keep it up if he's moved on?
I think many of us struggle with this, because once we move on (I mean really move on), then generally we wouldn't reach out to an ex for emotional support. But a pwBPD does not process relationships, grieve, or think in the way we do. The appearance of having "moved on" doesn't mean that feelings for previous partners don't come flooding back... .and then disappear or transform into the opposite shortly afterward. It's part of the disorder, and is especially likely when a pwBPD is feeling fearful about engulfment/abandonment with someone new. I know how confusing and painful it is—I've been there, too.
In my case, pwBPD reached out after our breakup during times of increased intimacy in his current relationship. I reckon the fear and his need for soothing pushed him to look for emotional support from someone who used to provide that for him, just in case all hell broke loose in his current relationship. Even though I wouldn't do that, it makes sense to me, for someone who is feeling such a high degree of fear.
Don't worry about the googling. It happens. Take the information (the feelings in your gut) from that experience and use it to gauge your stage of detachment. Most of all, be compassionate and gentle with yourself. Healing takes time, and slip-ups are a part of that—it doesn't mean you've lost ground AT ALL.
heartandwhole