Hi Lorrie,
It doesn't sound like you're doing anything illegal. It's perfectly legal to enter the UK multiple times and it doesn't sound like you've overstayed as a tourist. Be aware that entering the country frequently can arouse the suspicion of immigration officers and they can give you a hard time and/or refuse to let you in if they think that you're trying to settle here or think you won't be able to support yourself during your visit.
If you want to live here permanently, you'll need to go back to the US to apply. Since you haven't been living together two years, you won't qualify for the unmarried partners visa. You can apply as a fiancee' (you'll need to get married within 6 months of entering the UK and can't work until getting married and receiving your FLR), a spouse (you can get married in the US -- your partner won't need a visa -- and then apply for your spousal visa, which gives you the right to work right away), a student (must be enrolled in a UK university as a full-time student), a work permit (you'll need to find an employer to sponsor you), or a highly-skilled migrant (if you qualify).
It sounds like the fiancee' or spousal visas are really your best option if you aren't interested in a work permit and aren't a student.