You ask about taking control of mom's social security and medicare, due to her advanced brain tumor. I presume that she is no longer competent to communicate with the Social
Security Administration. If she were competent, she could call the nearest Social Security office and ask for a change of address herself.
One option for you is to use the website to locate your nearest Social Security office. Visit the office (can take some time waiting, as it is first-come, first-served) and request that you become your mother's representative payee. You will need all the documentation to prove that she is incompetent (letter from her doctor and/or medical record) so that checks will come to you as her representative. That process also requires giving a sworn statement, as well as other documentation verifying your own information as a legitimate person to act on her behalf. The Social Security Administration, itself, does not simply accept the usual power of attorney for you to work with them directly for your mother.
As representative payee, you can arrange for the direct deposit of your mother's checks into an account that you designate for her benefit. You will be a signatory on the account and can use it on her behalf.
Medicare is her health insurance benefit. You do not have to use a power of attorney for her to continue to access her benefits under Medicare. The program is run through the Social Security Administration. The change of address for her Social Security should be the record for where her Medicare statements are sent. The bill for her monthly payment into Medicare is normally deducted from her Social Security payment automatically. At around this time of the year, she will receive a summary of her benefit payments for the year.
As her financial power of attorney,you are responsible for seeing that her Medicare co-payment is correctly made,balances not covered by Medicare or supplemental insurance are paid, and that her supplemental insurance premium is paid when due. You will also need to review all medical bills to be sure she is not being overcharged or billed for things she did not receive.
The Medicare website is not particularly user friendly, but if you are patient, you can eventually find what you need. If you use the toll free number, call very early in the day or as late as they receive calls, due to very heavy use. If you do not get the help you need, ask to speak to a supervisor. They are supposed to be able to answer all the questions you raised here. Persist if you need more help. It can definitely try your patience, and not all SSA employees are helpful, but the competent people there do want to assist you.