You may need to adjust the hole size down a bit, the pins on these are IC size (which Fritzing uses a 0.035in hole for rather than 0.038in. As I recall (I don’t have one to hand) there is a larger lip at the top of the pin I think meant to press fit them in to a plated through hole. In your case where height matters you probably want the hole big enough to allow that lip to fit in it to minimize the height. I also had problems finding any on ebay, there used to be some listings with 10 each male and female 140 strips for I think about $5 (I bought some 5 or 10 years ago) but I can’t find such a listing now (which may just be wrong search terms.) when a single strip of 150 was $6 from digikey (to be fair, I think the digi ones were gold and the ebay tin, but still …)
Peter