Original

A tough day..

This morning, we have a lot of snow accumulated.
Since last night, the snow had been falling non-stop!
Besides, this snow was heavier than the meets the eye, because
it had much water more than usual. Of course the shoveling snow
became really a tough work. Juts more to say, someone who did
the shoveling snow was my father.. I am grateful to him.
He said that as the snow was heavier his waist was hurt..

Correction

A tough day.

This morning, we have a lot of snow accumulated.
Since last night, the snow has been falling non-stop!
And this snow is heavier than the meets the eye, because
it is wetter  had much water more than usual . Then of course,  the shoveling snow
becomes really a tough work. And  just  more to say more,  the someone who did
the shoveling snow was my father.. I am grateful to him.
He said that as the snow was so heavy heavier his waist was hurt.

Notes: Some of your phrases are different to those that a native English speaker would use.

They are not incorrect and they would certainly be perfectly understandable. They are just slightly unusual.

In fact, I was thinking that perhaps I did not need to correct the sentences.

What I have done is

. altered the word  so the snow has been falling which means the snow started falling last night and continued.

. much more water has been replaced by wetter

. I have changed Juts more to say to And just to say more. That corrected a spelling mistake juts and put in the normal order that an English speaker would do.

. I have added the in front of  someone  this gives a sense of being special to the person being described.

. I have removed the word was from was hurt as that is the common usage.