Here is a direct way to see the answer is 'no'.

Let our countable set be the set of all integers $\mathbb{Z}$.

What is the sign of $(12)(34)(56)(78)\dots$?(1,2)(3,4)(5,6)(7,8)\dots$? (Note that we're fixing all nonpositive integers here.)

Whatever it is, you can multiply by the transposition $(12)$ (1,2)$ to get a permutation with the opposite sign, then you can conjugateby $(1357\dots)(2468\dots)$, , which doesn't affect the sign, but gives you by $(\dots,-3,-2,-1,0,1,2,3,\dots)^{-2}$ giving back the element you started with. Contradiction.

Here is a direct way to see the answer is 'no'.

What is the sign of $(12)(34)(56)(78)\dots$?

Whatever it is, you can multiply by the transposition $(12)$ to get a permutation with the opposite sign, then you can conjugate by $(1357\dots)(2468\dots)$, which doesn't affect the sign, but gives you back the element you started with. Contradiction.