I suppose that you meant the mechanism I attached, but I don't know how the second step happens. Maybe aldol addition?
No, there is no aldol in the isomerisation, just carbonyl-enol equilibria.
Let's try working from both ends. You have drawn 1,3-dihydroxypropanone and its enolate, fine. We will call that enolate 1.
Now draw 2,3-dihydroxypropanal, and draw the enolate you would get from that by deprotonation. We will call that enolate 2.
How are enolates 1 ad 2 related and how might they interconvert? There is an equilibrium between them: