Should I include Box 17 from my W-2 for my "Total of all your payments and withholding" in the State Tax Refund section (1099-G)?

Yes, box 17 on a W-2 reports state tax withheld and paid. 

You may have paid other taxes as well, beside through your pay. 

However, if you took the Standard Deduction on last year's federal return, the state refund is not taxable and you needn't enter it.

When a person Itemizes on their federal return, they deduct state tax paid. If they later get a state refund, they might need to claim the refund as income. 

When a person takes the Standard Deduction, state tax was not an issue on the previous federal return, and the refund is not reported.

Even if a person Itemized, and deduced state income tax paid, if the state tax was limited because of SALT, any refund over that amount would not be reported.

(Taxpayer paid 12,000 state tax but could only claim 10,000. Any state refund 2,000 or less needn't be claimed) 

Or if the itemized deductions were only slightly more than the standard deduction, only part of the refund would count. 

(Taxpayer claims 7,000 state tax paid and gets 1,000 refund, however their itemized deductions were only 400 more than the standard deduction, so only 400 of the refund is claimed) 

You don't need to do the math, you can have TurboTax figure it out for you but always good to know how taxes, and refunds, work.