Why does my tax liability increase when I try to claim my 27 year old son who lived with me and earned 700.00 for year, he got a 1095 from but it’s going off my income

Whether he files first or not, will not have any affect on who has to enter the 1095-A on their return.  If you claim him as a dependent, then he would select that someone else is claiming him, then he would NOT enter the 1095 on his return. The Premium Tax Credit is based on the income on the tax return.  So, if you are claiming your son, then his Premium Tax Credit would be based on your income not his, as they are looking at it like you are then responsible for his health insurance since he is dependent on you for financial support.

Since your son is 27 and your tax liability is going up, you may leave him off of your return if you choose. If you do not claim him, then he WOULD enter the 1095-A on his return. 

 The benefits to claiming a 27 year old would be the $500 Other Dependent Credit and possibly filing as Head of Household.  The HOH status would only apply if you were otherwise filing as single, it would increase your standard deduction from $14,600 to $21,900.  If you are married and lived with your spouse during any part of the last 6 months of the year, you are not eligible to claim Head of Household.