Is this paragraph grammatically correct? If not, please correct me plz?

Grammatically, it's "pride myself on", not "in". However, you might want to reconsider the use of this phrase. They won't be too interested in your pride. Just say you can work to tight deadlines.

You need to research the name of the actual person to whom you are directing your letter. "Dear Sir" implies you were too lazy to look it up. Rather than vaguely saying "in this regard", be more specific about what the job is that you are applying for, and also what applicable skills you will bring to it. Are you hoping to work from home or simply obtain a contract position? Also, eliminate the unnecessary word "extremely" in the last paragraph.

The real problem with this letter is that you are basically talking about what this company can do for you, rather than what you can do for this company. Every sentence begins with the word "I". An opportunity for YOU. YOUR pride. YOU are "very confident and highly motivated". YOUR "wealth of experience". [The company would surely prefer to hear about years of experience, how recent, and in what capacity.] It sounds like inappropriate bragging when a simple statement of your qualifications would be more to the point. They will judge your confidence level and motivation when you go in for an interview.

Finally, the closing "faithfully yours" went out of style about half a century ago, and for good reason. You don't know them and they don't know you, so why would you describe yourself [in Dickensian fashion] as "faithful"? A simple "sincerely" is more up-to-date and believable.

There are many sample cover letters online which can help you rewrite yours. Good luck!

It is grammatically correct but sounds too desperate and obsequious to be taken seriously by any employer.

It is grammatically correct. At the end of the first sentence, the phrase "in this regard" sounds a bit odd. It is understandable, but it doesn't sound like something a native speaker would say. You might consider something like "...would be delighted to be considered for any freelance or contract vacancy you seek to fill."