I suppose you know that this process you're looking at is very similar to the one treated in Boris Pittel's paper "On Spreading a Rumor" (SIAM Journal on Applied Mathematics, Vol. 47, No. 1, Feb., 1987). In your notation, I believe his process would be written $$A(t+1) \sim A(t) + Bin(n-A(t),1-(1-1/n)^{A(t)}).$$ So I would recommend knocking on his door and seeing what he suggests!
I suppose you know that this process you're looking at is very similar to the one treated in Boris Pittel's paper "On Spreading a Rumor" (SIAM Journal on Applied Mathematics, Vol. 47, No. 1, Feb., 1987). In your notation, I believe his process would be written $$A(t+1) \sim Bin(n-A(t),1-(1-1/n)^{A(t)}).$$ So I would recommend knocking on his door and seeing what he suggests!