As Geeta has said, unfortunately options for foreign students to stay in the UK are very limited now. If you're ready for marriage, that's going to be your best option, as there isn't a way to get a visa to extend your stay after being a student. Tier 2 is technically an option (we've been told here to never say things are impossible when giving immigration advice

), but the chances of finding an employer to sponsor you are very, very slim right now, as there are many British recent grads struggling to find work (10,000 recent grads applying for ten places in a graduate scheme isn't uncommon) and Tier 2 is really for skilled migrants with work experience. Even companies that employ foreign students through internships (paid or not) aren't able to hire them to stay on after they graduate.
Someone else will need to clarify whether or not you can apply for your fiance visa in the UK or if you'll need to apply from outside the country. There are a lot of rules when it comes to switching visas inside the UK, and I'm not up to date on those rules. I will advise that it is usually easier to fly to the US and get married, apply for your spouse visa and then move back to the UK. Your fiance will not have to get a visa to get married in the US, but you will need a fiance visa to get married in the UK. So marrying in the US will save you the cost of a fiance visa and could end up being cheaper depending on flight costs.
Hope that helps.