Yes, the income from the severance package is taxable income and this will impact the amount of the stimulus to be received.

It appears the new 2021 stimulus will work the same as the old 2020 stimulus did.  If you do not get the full amount in 2021, you will get the rest when you file your 2021 return, in 2022. The amount you get will be based on your 2021 income, not your 2020 income. 

Reference: https://www.kiplinger.com/taxes/602381/the-current-plan-for-a-1400-third-stimulus-check

"As with the first two stimulus payments, your third stimulus check would be an advance payment of the recovery rebate tax credit for the 2021 tax year."