I am feeling cold.

[bebe heart]

It's very cold today and I am feeling cold.

It's winter, the weather is cold.
Are the above sentences correct?

 

[emsr2d2]

It's very cold today and I am feeling cold.
It's grammatically correct but there's no need for "feeling". You can just say "I am cold".

It's winter, the weather is cold.
You need to either use a semi-colon between the two parts or separate them into two sentences.

Are the above sentences correct?

See above in red.

 

[Tdol]

If you don't want to repeat the word cold in the first, you could replace very cold with something like freezing.