This is what I've found in Collins dictionary: a variant (now rare) spelling of dote :-)

"Doat" means to do lots of kind things for a person. It will be used often for when a person is sick and needs extra love shown to them. So, someone will not only make food for them or give them their medicine, but also do extra things like buy them a favorite treat or rub their feet in order to help them feel emotionally better, as well as physically. It can also be used in other instances. Another way it is often used is in romance. A person will "doat" on the person they are falling in love with.

Is it from an old book?