IRS withholding calculator says we're holding back more than enough, but we appear to owe $4000 in federal taxes, but getting $2000 back from State. What's going on?

I think you mean that the IRS calculation is on the W-4 that you gave to your employer.

If you've prepared a W-4 and still owe substantial taxes, you may have misunderstood how to post the form.

If you have multiple incomes, your taxes will go up rapidly.

Taxes start at 10% and go up to 37%.  I'll show fewer tax brackets for simplicity.

For Married Filing Jointly

$0 to $22,000 - 10%

$22,001 to $89,450 - 12%

$89,451 to $190,751 - 22%.22

If your married and both of you work.

You both have taxable income of $50,000.

You get taxed at 10% for $22,000 = $ 2,200

And at 12% for the next $30,000 =  $3,600

Your spouse is taxed 12% in $37,450 = $4,497  (the balance in the 12% bracket

Finally she will pay 22% on the last $12,550 = $2,761

For a total tax of $13,048

The W-4 has to show how the income accumulates, and must be complected for all sources of income.