The F-4E was available in 1967 with the internal gun and 4 sparrow missiles vs Mig-19 with no missiles.
North Vietnam's Air Force used the MiG-19 much later in the air war than the MiG-17 and the MiG-21. MiG-19s, despite their limited numbers, were involved in extensive combat during Operations Linebacker 1 and Linebacker 2 (aka the Christmas Bombing). The NVAF claimed only seven victories using the MiG-19 over US aircraft, all of which were F-4 Phantom IIs.[4] Primarily because of the aircraft's twin engines, which created a maintenance nightmare, the MiG-19 wasn't favored by North Vietnamese pilots. While the MiG-17 had maneuverability and the MiG-21 had speed, the MiG-19 had a combination of both, but not to the same degree as the others. North Vietnam used the MiG-19 from 1969 until the 1980s when it was replaced by newer aircraft.
Compared to the F-4 Phantom II however, although lacking mounts for air-to-air missiles, it had the one advantage that the early model Phantoms did not have: it was armed with cannons. Confirmed aerial victories by MiG-19s while assigned to the 925th FR, which match US records occurred on: 10 May 1972 in which two F-4 Phantoms were shot down by MiG-19s flown by Pham Hung Son and Nguyen Manh Tung. Both NVAF victories over the F-4s were accomplished by cannon fire, pilot Nguyen Tung's downed USAF F-4 Phantom was manned by Major Robert Lodge and his WSO 1st Lt Roger Locher.[5][6][7] Combat results of the 925th FR using MiG-19s, according to the North Vietnamese Air Force were: two F-4s on 8 May 1972 (MiG-19 pilots: Nguyen Ngoc Tiep and Nguyen Hong Son); two F-4s on 10 May 1972 (MiG-19 pilots (both previously mentioned): Pham Hung Son and Nguyen Manh Tung); one F-4 on 18 May 1972; and two F-4s shot down on 23 May 1972 (MiG-19 pilots: Nguyen Hung Son and Pham Hung Son);[4] these losses were in exchange for 10 MiG-19s lost in aerial combat with US jets. The MiG-19 did make history in one manner however; on 2 June 1972 over the skies of North Vietnam, the MiG-19 has the inauspicious honor of being the only recorded jet fighter[citation needed] to be shot down in aerial combat by cannon fire at supersonic speeds, by a USAF F-4 Phantom piloted by Major Phil Handley.
You were correct about the MiG-19 (J-6) that the NVAF used lacked A2A missiles. The article states that they didn't even have the mounts. Up until about 1972 the MiG-19 gave better than it got against the F-4 Phantom. It really wasn't until US pilots trained at "Red Flag" in dissimilar combat began using their Phantoms to best effect.