You do what ever good for you and your family

If this position is better than the offer you got, and could give you better salary, you can choose to tell them that you have an offer.

If they like you, they could expedite the offer process. (One time, I had to call our headquarter to expedite process to get this guy to US since he already had an offer from a German company).

Also if you got a better offer when you already start to work, quit and go to the better place. (This happens all the time in our company. I had one lady came in first day and quit since she doesn't like the office space. lol...). So don't be shy.

As long as you have skill and talent, you can make the choice.