I made the mistake of giving a salary number first. Now the company has given me an offer letter. How do I counter without sounding cheap or stubborn?

I already work for this company in an hourly position. A salary position opened up, and I applied for it. During the interview, I was asked how much I expected to make. I panicked and gave a number(I've always been told not to be the first to give a number.)

Now the company has given me an offer letter. They have offered 5k less than what I asked. The thing is, with overtime, I don't make much less than what they offered right now. Should I be firm in my first number? Or should I ask for 3-4k more?

Also, this is a weird question, but it has me just as anxious as the salary thing - I was handed an offer letter by the GM. The offer letter is signed by the HR rep. To whom do I address my counter offer?

Thanks in advance.

Some possibly relevant details:

I currently work in a different department, but have slightly more than beginner's knowledge of the new job.

Both the salary I asked for and the salary offered are, from what I can tell, average for someone in this position/industry.

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