[talanum1 (OP)]

When a particle and its antiparticle meets at exactly equal and opposite momentum, does its annihilation still produce a photon?

[Bored chemist]

Yes and no.
Annihilation, as always, produces (at least) a pair of photons.

From some point of view, every annihilation happens when "a particle and its antiparticle meets at exactly equal and opposite momentum".

Imagine you were sat on the line between the two particles.
They would both be heading directly towards you.

Now imagine you were on that line but moving along it at the average of the speeds of the two particles.
They would come directly towards you at the same speed.
So their total momentum, from your point of  view (and that of the bit of space where they collide) is always zero.