Oh man, bummer. I've only ever had one show up on nearby/sightings, and it was when I was traveling out of the country, back when spawns only lasted 15 minutes. When I saw it, I had to wait for the waitress to bring me my bar tab, and then I ran in every direction trying to find it, but it despawned before I could. The VERY NEXT DAY, they introduced the pokestop-based tracker which told you exactly where to go to find a mon.
I wish the three steps were still there
Answers
While there may be a placebo effect to this theory, something I appear to have had some success with is switching to a different level ball after several failed attempts, my thought is that it confuses the system's calculation of the catch rate. I notice particular success with common mons that break free of multiple poke or great balls in a row when I switch to the other, and have even had some success with the rare wild dragonite or alakazam by switching to a great ball or two after they break out of 5-10 ultras in a row.
For an instance like that you may wish to consider downloading niantic's other game Ingress. You don't actually have to play, but it can occasionally be useful for tracking a rare mon from the Nearby. There is a resource in it called XM that appears on the in-game map as clusters of blue/white dots, and where large clusters appear there are often spawn points for the rarer pokemon in Go.