God did not allow Moses to avoid his responsibility. However, he responded to the concern of Moses by appointing his brother Aaron to speak for him (14:16). Aaron would serve as the mouthpiece of Moses before Pharaoh and the people.
As the story continues, the text records fewer occasions when Aaron speaks on behalf of Moses. Moses increasingly speaks directly.
This suggests that the speaking difficulty of Moses may not have been as severe as he feared. It may also be that his confidence and ability grew with experience.