Right Name wrong Person

In April I moved to another state to live and for work, I bought a house with 3 bedrooms 2 baths and just under an acre of land. I'm Darrel 29 and once I moved in and settled into a routine, I'm not a party guy I don't go out much. I got a letter addressed to me but didn't know the sender's name. But I replied and then received another asking to visit me and talk about old times. I agreed to meet the person and a week later she turned up, Cathy 27 the same last name as me and good looking I think. It turns out that Cathy is looking for her cousin of the same name as me. He is her only living family member she knows of since her mother passed away in March. My name had popped up on a website and she thought it was her cousin she was looking for. She was only able to contact me through my home address as I had registered for voting in my new district. I only put my home address and no phone as I didn't want phone calls from the candidates during elections. As Cathy was in town for a few days and was going to continue looking for her lost cousin. I said I would help as much as I could, after a few days and no luck, she had no job to return too and she started to look for work locally as this town was the last place her cousin was known to have been. I offered her a room if she wanted it and she accepted, she quickly got a job and we settled into a friendly relationship. She a very good cook and does most of the cooking and that is how it was for a month or so. In June she found her cousin had moved aboard to work and tried to contact him through the company he was to be working for. Only to receive the news that he was dead in an accident on the oil rig he was working on. They had tried to find his family but they didn't have any contact information and they had advised all the people in his cellphone contact files. This left Cathy very sad and I tried to cheer her up, I took her out for a nice meal and a movie and a few drinks and once back home, we were both a little drunk and I kissed her and we slept together that night and had sex again in the morning. After we were a little shocked by how easily we had fallen into a sexual relationship. She said She knew I was drunk and she doesn't expect me to continue with a relationship started by us drinking too much. I also had been thinking and I replied saying I had enjoyed the sex and her company and if she would like too, I would like to have a go at a relationship with her and see how far it goes, also saying she didn't have to sleep with me again unless she wanted too. She smiled at me and said she would think about it. A week later and having not heard from about a possible relationship. I decided to ask her out again and hoped that she would agree. While drying the dinner dishes with her I asked, she looked at me and said okay. We have since been dating regularly and have also started to sleep together every night.