In general, if you prepare the W-4 correctly, this will be taken into account, at least as far as wages is concerned. When changing jobs, or adding a new job, you should prepare new W-4s for all the jobs. Follow all the instructions and include all the worksheets on all 4 pages of the form, including the worksheet for multiple jobs. Use the results of the calculation to update your W-4s at all your jobs. Or, use the IRS online calculator to determine what to put on the W-4s.
https://www.irs.gov/individuals/tax-withholding-estimator
For the annuity, you can generally have any amount withheld that you want, you just have to tell the payer. If you know your overall income will be in the 22% bracket, ask for 22% withholding.
(In your example, $120K of wage income minus the standard deduction results in $90,800 of taxable income, so that's all in the 12% bracket. If you then add another $26K on top, most of that will be in the 22% bracket. So you need to tell the bank or broker to withhold that much.)
