Which job types are eligible for a H1-B visa holder

I was staying in US on H4 visa. Got H1-B Visa through a consultancy this year and got eligiblity for working in US from October. But I could not get placed and have not started working and am still searching vigorously. I am only searching for C2C since I was told i am not eligible for other job types.
Firstly, Can you tell me which job types can be tried by only a H1 visa holder and what problems will be faced if any Fulltime job positions are applied?

Secondly, Also I was told H1 status will be revoked if I could not get a job immediately since its already December . Can you suggest on this too?