I have just come home from grocery SHOPPING.I was stuck in THE rain and got a bit cold as I was wearing shorts and a t shirt.
The only part of the statement missing is 'the'. The rest is correct. But If i said this I would say:
I'm just home from (shopping at) the grocery store. I got caught in the rain though and got a bit cold as i was only wearing shorts and a t-shirt.
yours is correct but that sounds a bit more natural to me. 'I'm' means 'I am'. It's just the shortened form and rolls off the tongue smoother.