If I cash out an annuity and have them withhold the 10% for federal taxes at the time of surrender, will I have to pay taxes on that money again when I file taxes?

Yes.  They are only withholding the 10% penalty, not the income tax that needs to be paid as well.

Also, whenever you have income and you have taxes withheld, when you file your taxes at the end of the year, it is basically a reconciliation of what has been paid and what is owed.  So if you have enough withheld on the annuity (income taxes PLUS the penalty) then you would not necessarily need to pay more, but it would all be put in one pile of income and taxes withheld to determine if you owe more or will get a refund.

You can do an estimate and have additional taxes withheld. The amount will be based on your overall total income and your tax bracket.