How to choose a distribution for the likelihood step?

Hi,
I have tried with halfnormal and gamma both.The posterior statistics is very close.
I am trying to see posterior predictive.

I am sincerely thankful for your responses.

May I have one more query? I am trying to plot the joint distribution of my mean and sigma. But I am not successful yet. Can you kindly advise? A sample code will be highly appreciated.

Thanks in advance
Best regards