Any corrections, please~

There is still a heavy fog today.
I should go (or "I must go" which introduces more of an immediate urgency) to work soon; otherwise, I would will be late.  ("will" is future tense; whereas, would is normally used for past tense). 

Thankyou for reading and correcting.

Any corrections, please~

There was still a heavy fog today.
I am going to work soon, otherwise I would be late.

Thankyou for reading and correction!

The fog is just as heavy as it was yesterday. I have to leave for work soon, otherwise I will be late.

Any corrections? Please advise, thanking you in anticipation.

Any corrections, please~

There is still a heavy fog today.
I should go to work soon, otherwise I would be late.

You're using still, so it's happening at the moment.