It is your responsibility to be aware of your financial obligations and contracts you sign.
It is not the responsibility of others to inform you of things you should be aware of. Your landlord is not required to inform you of something that is in your lease you signed.
Your lease agreement clearly state the date you lease would end. Your lease also state the requirements you would need to do prior to your lease expiring.
Since you failed to inform your landlord of your intentions to sign a new lease or to move, your landlord indicated that they were willing to continue to have you as a tenant on a month to month tenancy and there would be an increase in your rent. This is normal,they have indicated the conditions of you remaining in the rental.
You failed to indicate your desire as to sign another lease or move.
You took it upon yourself to move once your landlord indicated the terms for you to remain a tenant. Again you failed to provide the proper notification that you would be moving.
You want your landlord to comply with various rules in this situation, yet you have failed to comply with any of the rules of your lease or state requirements. In your state, without a lease, you are normally required to provide your landlord with a 30 day notice that you would be moving. Apparently you failed to do this requirement.
If you have a feeling that you were not legally given the proper notice to move and you have the legal grounds, by all means you would want to take your landlord to court and sue them. You should keep in mind that you do not have clean hands. You have not complied with providing your landlord the 30 or 60 day advance notice that you would be moving.
Since you failed to do this requirement you could be required to pay the 2 months rent, lose a portion of your deposit for damage to your formal rental. Your feeling about this matter would not matter. You feelings have no legal standings in any court.
The fact that you are young and have a child that is in need of special attention is not a legal defense in court. Also ignorance of the law or your lease agreement is not a legal defense in court.
You are bringing up things about your landlord and the way they have conducted their business, as a means of supporting your position. This things should have been taken care of as they happened.
I hope this has been of some benefit to you, good luck.
"FIGHT ON"