بسم الله الرحمن الرحيم
Yes, in case your wudhu breaks you will have to make fresh wudhu and will have to wash your feet after putting off the shoes and socks as it is fardh to wash the feet and it is proved from the Holy Quran.
Socks means one which is made of leather and called Khuffain. There is no need to put it off as it is enough to do masah over it, but it is not lawful to do masah over the prevalent nylon socks or cotton socks (whether thin or thick), and if done the wudhu shall not be valid.
Allah knows Best!
Darul Uloom Deoband