This could be construed as a retaliatory eviction, but the only way to really find out is to continue to pay rent, not move out, and then go through the process.
Hopefully you can reason with your landlord and try to get the extension that you need. If the landlord is halfway reasonable, he should allow you the time, especially if you consent to the rental increase that he wanted.
Most single family homes are exempted from any kind of rent control, and you are correct that if the landlord wants to increase the rent more than 10%, you need 60 days notice.
You can also sweeten the deal by telling the landlord that you will cooperate with any new prospective tenants.
You do not want any kind of UD filing on your record, especially if it goes to judgment. That could affect your credit for up to 10 years and could make it difficult to buy a house, or refinance in the future.
So do your best to work it out with the landlord if you can.
If it goes to court, your payment record, as well as the fact that you exercised your rights to a lawful rent increase, and then an eviction, may go your way in court.
This is general legal advice intended for informational purposes only and does not create and attorney/client relationship. If you wish further advice, please contact an attorney of your own choosing or you may contact me for further advice and make other arrangements including retaining my services.