Should we Refinance our house?

We are looking at refinancing out mortage from a 30 yr to a 12 or 15 yr loan. My question is they want to charge about $ 14,000 to refinance here in FL and our mortgage would go up $ 600 more a month. If we just sent in $ 600 more a month without refinancing would it be the same as refinancing or would we actually have to refi in order to save as if we did have a 15 year loan? Sorry if it’s confusing.

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