If you want to work for that employer then the only option is sponsorship. You cannot extend your PSW visa. If your visa expires then you must stop working immediately and you must either leave the UK or submit an application within 28 days. If you stay longer then you seriously risk causing problems with future applications. You should see if your employer is willing and able to sponsor a Tier 2 General visa for you.
My current employers are willing to sponsor me for a Tier 2 visa, but I've also been offered a job at a different firm, so I will likely change jobs by mid-December beginning of January. This is why I'm trying to avoid getting my current employer to sponsor me since I will change companies only a couple of weeks after. Ideally there would be transitional solution. What I would like to know is:
If I apply and change my visa status to tier 2 sponsored by my current employer, how hard would it be to switch to the other employer a few weeks after? Do I have to apply all over again with the new employer or is there an easy way to do it?
Are there any rules against me applying for a tier 2 visa twice for different employers in such a short time?