Cohlmia had the lead after Monday's opening round of stroke play with a 5-under 67 and followed it up with a 2-under 70 at the Gold Mountain Golf Club's Olympic Course for a two-day total of 7-under 137.
After bogeys on two of his first three holes Tuesday, he rallied with four birdies the rest of the way.
'I didn't short-side myself nearly at all this week, at least so far, which makes it easy to get up and down,' said Cohlmia about minimizing his mistakes. 'Not a lot of pressure putts. It seemed like if I missed the green, it was a pretty easy chip -- I chipped it to a couple feet and made it.'
Metzger, meanwhile, carded a round of 4-under 68 on Tuesday after opening with a 69 Monday.
Now it's down to match play, as the top 64 players from two rounds of stroke play qualified for Wednesday's action.
Martin Ureta of Chile, last year's Public Links runner-up, finished third in stroke play with a 6-under 138 after his second straight round of 3-under 69.
'I'm in a good position for match play,' said Ureta. 'I putted very well and that's good for match play on this golf course.'
Defending champion Clay Ogden of West Point, Utah tied for 12th at 1-under 143.
A 10-man playoff was needed to fill the final eight spots in the field of 64 and the odd men out were Nicholas Van Vuuren of South Africa and Tyler Trout of Leawood, Kansas.
One of the oldest United States Golf Association events, the U.S. Amateur Public Links Championship is being held for the 81st time. Some of the past winners who have gone on to have professional careers are: Jodie Mudd, Billy Mayfair, Trevor Immelman, David Berganio, Jr. and 2004 champion Ryan Moore.
Immelman picked up his first PGA TOUR win this past weekend at the Western Open, beating Tiger Woods and Mathew Goggin by two shots.