Why do the dry and moist adiabatic lapse rates converge with height?
If you look closely at the diagram you'll notice height isn't the primary cause for the convergence between the adiabats, though it is the driver. 

Note that toward the upper right there is still appreciable difference between the adiabats and as you move to the left the height they converge at decreases. 

The reason for this, as your intuition suggests, is that condensation ceases. The reason, simply, is that there is no more water left in the parcel to condense and release latent heat. 

Recall the lines of constant water vapor mixing ratio (the lines you use to connect surface dewpoint to the LCL). These lines decrease to toward the left and are sloped steeper than the isotherms. As moist parcels ascend the latent heat release is correlated to decreasing water vapor. At some point we exhaust the water vapor and any further ascent cools at the dry adiabatic lapse rate as there is no more condensation. This is what causes the moist and dry adiabats to become parallel at upper levels (higher for warmer / moist surface conditions and lower for cooler / drier source parcels). 

In the comments, concern was expressed that the skew-t isn't capturing the relevant physics, including how a parcel runs out of water vapor.  The skew-t does indeed capture this information and one way you can observe this is by calculating equivalent potential temperature, $\theta_e$.  This temperature is the temperature of a parcel with all of the water vapor condensed out of it. The more moisture in a parcel, the higher the equivalent temperature will be over its actual temperature.  To calculate it on a Skew-T you raise a parcel high enough for the moist and dry adiabats to be parallel, then descend the parcel.  On the way up once saturation is reached you'll ascend via moist adiabat and once the water vapor is exhausted you descend via a dry adiabat.  You can try this for various source parcels and convince yourself that the Skew-T captures these physics.  You should note that the more moist a surface parcel is, the higher it must ascend and the warmer the equivalent temperature will be.