I think you're computing your taxes incorrectly, or are you also factoring state taxes? Here's what I get for federal only, assuming all 3 kids are under 17 for the "child tax credit" portion:
"Lots of Overtime"
Ordinary Income: $125,000.00
Deductions/Exemptions: $ 56,700.00
Taxable Income: $ 68,300.00
10% on first $18,650: $ 1,865.00
15% on remaining $49,650: $ 7,447.50
Total Tax on Ordinary Income: $ 9,312.50
Ordinary Taxable Income: $ 68,300.00
Long Term Cap. Gain: $ 15,000.00
Total Taxable Income: $ 83,300.00
Less Top of 15% bracket: $ 75,900.00
Portion of LTCG over: $ 7,400.00
15% on $7,400 of LTCG: $ 1,110.00
Tax on Ordinary Income: $ 9,312.50
Total Taxes: $ 10,422.50
Less Child Tax Credit: $ 2,650.00
Total Tax Owed: $ 7,772.50
"Less Overtime"
Ordinary Income: $117,600.00
Deductions/Exemptions: $ 56,700.00
Taxable Income: $ 60,900.00
10% on first $18,650: $ 1,865.00
15% on remaining $42,250: $ 6,337.50
Total Tax on Ordinary Income: $ 8,202.50
Ordinary Taxable Income: $ 60,900.00
Long Term Cap. Gain: $ 15,000.00
Total Taxable Income: $ 75,900.00
Less Top of 15% bracket: $ 75,900.00
Portion of LTCG over: $ 0.00
15% on $0 of LTCG: $ 0.00
Tax on Ordinary Income: $ 8,202.50
Total Taxes: $ 8,202.50
Less Child Tax Credit: $ 3,000.00
Total Tax Owed: $ 5,202.50
So on your $7400 in additional earnings, you pay $2,570 in additional tax - netting you (ignoring payroll taxes and any state income taxes) $4,830 or $24.64 / hour for your 196 hours.
But what you do with the extra money matters - you didn't mention any IRAs - if you fund an IRA for your spouse (this reads like a single-income scenario), that's another $5500 of ordinary income sheltered. You have a 457 - do you also have access to a 403B or 401K? You can do both a 457 and a 401K/403B in the same year - that's another $18K you could potentially shelter, and if you do, you'll likely be able to deduct a traditional IRA for yourself - another $5500. Do you pay any health, dental, vision insurance or similar via your paycheck? Maybe you have some sort of pension that gets deducted? Those also shelter you from income tax.