yes, while the wet bulb depression (the difference between the dry-bulb temperature(Air temperature) and the wet-bulb temperature)comes to close , the rh will increase, and vice versa
The answer is both no and partly yes. First of all, if you need to reduce relative humidity, you need to cool air and take its water content. Therefore, you decrease DB. However, when you finished to decrease water content, then you can heat air to the inital DB. Then, you have DB unchanged and Rh decreased. Please try to draw process in psychrometric chart. It had better to do it for understanding.
Typically in a cooling coil, we do cooling and dehumidification, which means reduction of Dry Bulb and RH. However, if in the process of dehumidification, we end up over-cooling, then a sensible heat addition is done to bring back the temperature of air to the desired level.