I heard that there are some visa agents who can get F visas for louwai like you,ie those who are not attached to companies;but they charge a pretty penny for their services.Of course,said visas have been obtained by using special relationship.That's why they are costly.
Where I'm teaching at present,there was an old FT accompanied by her husband.He was in ill health.So the local FAO refused to extend his visa.However,his wife managed to get him an F visa through a Shanghai visa agent so that he could stay with her in China.Unfortunately,that couple are not here now.So I can't find out more about what she did.I recall her telling me it was expensive.
The good thing about having an F is that you don't have an accompanying residence permit.So you can move around China freely,teaching immediately you have found employment.