How do I complete part-year resident state returns when each of them wants me to enter the Adjusted Gross Income for the other state before finishing?

TC-40s is for credit paid to other states. it's rare,( but not unheard of  - cases like rental property or income from certain businesses)  when a part-year resident has income taxed by both states. if you have no items of AGI taxed by both states, you should not be completing this form.

if you do have such a situation then the number it's looking for is the amounts on lines 1 to 11 of the federal 1040 taxed by both states.  

From Utah instructions

Part-year residents rarely qualify for this credit. If you are domiciled in Utah for part of the year and domiciled in another state for part of the year, you may only claim credit on the portion of income:

1. taxable in Utah,
2. taxed also by the other state(s), and
3. included in “Column A – Utah” income on form TC-40B.

Part-year residents must prorate and enter the tax paid to the other state(s) on TC-40S, line 6. The credit only applies to tax paid on the part of your income (TC-40, line 1) taxed by both states