Workers who hold existing long-term care insurance or purchase a new private policy before November 1, 2021 can apply for an exemption from WA Cares coverage. If approved, your exemption will mean that you aren’t required to pay premiums and will be permanently excluded from WA Care benefits

The IRS allows you to deduct state and local income tax or state and local sales tax as an itemized deduction, but not both.  It is not clear to me if this would even be considered an income tax under federal law.  If it was, you would still probably get a larger deduction from sales tax.

Oregon allows an itemized deduction for state and local income taxes you paid (not including Oregon state taxes and there is no option to deduct sales tax instead).  Oregon also has a credit for state income tax you pay in another state for income that Oregon also taxes.  However, again it is not clear to me if this would even be considered an income tax under Oregon law.  The WA Cares web site call them "premiums" even though they are collected in the same manner as taxes.

My guess is that Oregon and the IRS will take this position that this is not an "income tax" and therefore not deductible.  It would then have to be taken to court to see if a judge would call it a tax by another name.  You should check back as 2022 tax season approaches in the spring of 2023.   You could also list it as an income tax deduction on your 2022 tax return and maybe you will be the first test case.