I buy an OTM strike the price converge towards the selected strike price premium increase at a rate of delta does the rate of the increase slow down due to gamma?
If we assume constant volatility, gamma increases as the stock gets closer to the strike price. Thus, delta is increasing at a faster rate as the stock reaches closer to ITM because gamma is the derivative of delta. As the stock gets deeper ITM, the gamma will slow down as delta reaches 1 or -1 (depends if a call or a put). Thus, the value of the option will change depending upon the level of the delta. I am ignoring volatility and time for this description.
See this diagram from Investopedia: Gamma