Consider an entangled pair described by the wavefunction $$\lvert1,0\rangle = \frac{1}{\sqrt{2}}(\lvert\uparrow_1\downarrow_2\rangle-\lvert\downarrow_1\uparrow_2\rangle)$$ in in the $S_z$-basis. If the first measurement finds Alice's particle to be in the $\lvert\uparrow\rangle$ state, then Bob's particle is found to be in the $\lvert\downarrow\rangle$ state i.e. the entangled state above collapses to $\lvert\uparrow_1\downarrow_2\rangle$ which is not entangled. Since it is not entangled in the $S_z$ basis, it will also be unentangled in the $S_x$ basis.
Am I correct to conclude from this that an entangled pair becomes unentangled after the first measurement? Here, I am assuming that making a measurement of $S_z$ by Alice or Bob doesn't affect the space part of the wavefunction, which I am not sure though.