Your income was $1250 bi-weekly and now it is $1250 weekly - this is the reason for higher federal tax being taken out. As long as you are in lower tax brackets, you get taxed at lower rate. With income being almost doubled, you tax bracket is higher and hence, you owe more taxes. So higher withholding is required.
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Please note: This advice is provided with the understanding that all the relevant facts have been provided by you. Any change in facts might affect the advice given and hence may not be relied on in such cases.
The federal withholding of $118.79 is about the right amount. It seems that the earlier company was not taking the right amount of taxes from your paycheck for bi weekly payroll.