Multiple States - incorrect W2 information

I moved from Oregon to Hawaii in end of June 2021. I changed my residency to Hawaii and continue working for the same company as before. Unfortunately my employer did not switch my state withholding until October. Now while I'm working on my state taxes I assume I need to adjust the reported income. I calculated the correct income for each state and adjusted the state entries reported by W2 to proper, self calculated amounts. I've done this on both state returns. One value become lower while another one increased.

My Oregon return guides me through "Credit Tax Paid to Another State". It ends up being "$0" but I'm not sure if this is correct? In "Income Taxed by Hawaii" section I enter the following:

• Double-Taxed Income: 0
• Total Tax Paid to Hawaii: <value from my W2 for Hawaii>
• Total Income on Your Hawaii Return: <my self calculated adjusted income while in Hawaii>
  • I assume this needs to the same amount I am reporting on my HI return, but it is not the same amount originally reported by my W2.
  • Am I right here?

My Hawaii tax return asks me for "Wages and Salaries". I entered the self calculated amount adjusted to follow my residency status but different from the W2 reporting. Again, as I said before my employer start withholding my HI taxes in October. Now the most confusing part, the Hawaii State "Federal Schedule A - Taxes Paid" section. There are two values here which correspond to the W2 amounts which may have to be adjusted but I really do not follow this requirement. Here are the questions:

"Taxes on Out-of-State Income Earned While a Nonresident (such as tax withheld on an out-of-state job), and Real Property Taxes Paid on Property Located Out of State While a Hawaii Nonresident"

The default value here is the Hawaii "State Income Tax" as reported on W2. Should this be adjusted?

"Taxes on Hawaii Income, or on Income Earned While a Hawaii Resident, and Real Property Taxes Paid on Property Located in Hawaii or Paid on Property Located Out-of-State While You Were a Hawaii Resident"

The default value here is the Oregon "State Income Tax" as reported on W2. Should this be adjusted?

Next Turbo Tax leads me to "Other State Tax Credit Summary". Other State in this case is Oregon. For starters it is $0 but I go into it and provide details which ends up giving me a "credit amount".  Is this correct? 

In "Income Taxed by Oregon" section I enter the following:

• Income Earned in Oregon, and Subject to Hawaii Tax: <my self calculated adjusted income while in Oregon>
  • I assume this needs to the same amount I am reporting on my OR return, but it is not the same amount originally reported by my W2.
  • Am I right here?
• Tax Paid to Oregon: <amount originally reported on W2 for Oregon State>
  • Am I right here?

Here is another confusing section :-(, "State and Local Income Tax Deduction". The sections says:

"Hawaii requires you to subtract the state and local income taxes paid to this other state from your Hawaii itemized deductions amount if you are also claiming a tax credit for this state.

Your federal return included <$xxxx>  for ALL of your state and local income taxes paid.

For the state that you claimed a credit for taxes paid, subtract that state's income tax amount that you claimed as an itemized deduction on your federal return."

The amount defaulted here is my TOTAL (Hawaii and Oregon) State tax amount. Again I assume it needs to be adjusted by I'm not really sure about what is it asking for. Any ideas?

I know this was long post but the situation is quite complex and I wanted to explain it in details. I wonder if some of the experts out here can help me to resolve this puzzle. Thank You.