That is correct. However, we are talking about a higher level of absolute humidity (or constant relative humidity). That, by itself, doesn't require any extra energy. Now, it is possible (but not necessary) that this higher humidity also means a FASTER cycle of evaporation at sea level and condensation at higher altitudes (although that is by no means necessary to have higher absolute humidity by itself). This would then simply mean that convection, through this latent heat, has a bigger heat transport from the lower layers to the higher layers. However, convection will adapt perfectly to what is missing in heat transport to keep the (half-wet) lapse rate. If less transport is done through radiative transfer, more transport will be done by convection and vice versa, because the adiabat has to be restored.
So if a certain "matter flux" of convection is now more efficient in transporting heat (because containing more absolute humidity, and hence more latent heat), if convection has to transport the same amount of heat, the matter flux will diminish, to compensate (the air will rise SLOWER).
If the adiabat has to be restored, convection will compensate *perfectly* the "missing part" of the heat flux by radiation. The humidity level can (will) influence the adiabat and the lapse rate. But that, by itself, doesn't tell you anything about the amount of transported heat. And the humidity level by itself doesn't require any "extra energy".
You could picture this still differently. If you consider that humid air has more "latent heat" than "dry air" (which is the case), then this doesn't mean that HAVING more humid air requires a constant heat flux in. However, *making humid air go up and condense*, will require some heat flux up. How much, will depend entirely on the matter flux of that humid air up. That can adapt to the heat flux to be transported. And the regulatory mechanism is the lapse rate. If the humid air goes up too fast, lower layers will cool too much as compared to higher layers (where the heat is deposited), the lapse rate decreases, and convection slows down and stops. On the other hand, if there's not enough convection, the lower layers will accumulate heat, the lapse rate will increase, and convection is stimulated. Stationarity will set in when the lapse rate is given by the adiabat of the humid air.