Hi!
Are you able to get out and about in your city/town?
I have had good luck with finding a bank in my location. nobody would know I had another account unless I either told them or left paperwork around. You could choose a "paperless" option with them so you have no "paper trail" coming to you through the mail. Please DO check with any bank about their own policies re: Paperless options.
If you live in the U.S., you will receive a statement of any interest earned, which is to be included in your tax return; however, if that amount is low, you do not have to include it in your return. Please see the IRS guidelines on that amount. This statement is usually mailed to your home address.
Some banks allow deposits made by simply photographing a check, which may be a good option for you if you are depositing checks.
If you are looking to deal with well-established reputable banks, there are some which also allow online banking. I use USAA, which used to be only for members of the military and the Secret Service. It is now open to the general public. It also has ATM machines near me with no charge for use of ATM.
Please DO NOT take my recommendation blindly. Please do check out USAA and all other banks suggested to you.
You might want to look into ATM locations, account fees and any related fees when checking into any bank.
I hope this helps.