Does your credit score go down when you apply for a credit card

Does your score go down when you apply for an credit card. Im trying to rebuild my credit and I have one credit card Orchard Bank the limit is 300 dollars so do I pay 290 on it and leave the 10 dollars as the remaining balance how much lower could I pay on an balance of 300 for an score increase. Also I had an credit card by first premier that was closed and I owe 470 do I write then an letter like you have on your webpage or do I go ahead and call them and see what they say. I want to reopen with them again
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When you apply, you get an inquiry. Inquiries reduce your score by 1 - 6 points. So no.
Pay the cedit card off, completely and add more money to it to get a higher credit limit. ]
300 is considered and thin credit record.
Thickin the file by maying more than you charge to raise your limit.
Call first premier and reopen that account. Pay that one off and add to it like above.