The good news is that neither your income nor your assets would count toward your mother's eligibility for long-term care coverage in any state. For Medicaid medical coverage, Medicaid would
consider only the amount of direct support your mother actually receives from you. So, for example, if she received rent-free lodging and meals from you, Medicaid might count as part of her "income" the reasonable value of what she regularly receives from you. But even so, it would not count your income or assets. Since she will be paying you rent and paying for her own food, that won't be a problem in her case (unless she pays you considerably less than market value for rent).
Once your mother enters a long-term care residential facility -- nursing home or covered assisted living facility -- Medicaid will not consider anything she was receiving from you while she was living with you because she will no longer be receiving it once she enters the facility.
As for whether your mother can receive Medicaid coverage for assisted living in Washington or Georgia, you need to realize that there are restrictions on coverage in both states. In Georgia, the state Medicaid program can provide long-term care coverage, under a special "home and community-based services waiver" program, for residence in two different types of "personal care homes," which is the name used for assisted living-like facilities in Georgia. But Medicaid only covers care in certain types of homes, and only specific homes that are certified by and accept payment from the Medicaid program. If you are considering moving to Georgia and hope to have Medicaid pay for your mother's assisted living care, you had better make sure IN ADVANCE that there is a Medicaid-participating personal care home that she would want to live in that has a Medicaid-paid place for her (not all beds in a Medicaid-certified facility are Medicaid-paid beds; facilities often limit those to a certain percentage of their places).
Under the State of Washington's Medicaid program, a special "home and community-based services waiver" does cover assisted living and boarding homes (including for special dementia care), but only in those facilities that contract with Medicaid. Also, the amount that Medicaid will pay toward assisted living costs will vary depending on the needs, income, and assets of the resident. So, again, BEFORE you all move, make sure that your mother would have a place in one of these Medicaid-participating facilities, and that it is a facility that you and she would find acceptable.