Thanks for letting me know! This is much cheaper than consulting with a lawyer.
Is there a way to let employers know in a nice way that you would need them to sponsor you, but that you would do all the work and they just have to hire you? Sorry, I feel like now I should be switching to a different message board.
Well, the problem is that it's not quite as simple as 'just hiring you' - it is the company that has to do all the work, not you (there's not much you can do, to be honest, except wait for them to do the leg work and sort out your visa paperwork, so you can apply for the visa).
If the company is not yet a registered Tier 2 sponsor, they have to apply to become a sponsor and get a Tier 2 sponsor licence - this will cost them about £1,000 to do so, and currently sponsor licences are taking at least 5 months to be processed. Then after they have the licence, they have to give you a job offer and issue you with a Certificate of Sponsorship (CAS).
They can only get you a CAS if the job meets certain criteria... either if it is listed on the skills shortage list, or if you meet the criteria to forego the resident labour market test (i.e. due to studying for a degree in the UK), or they have meet the resident labour market test, proving they have advertised the job in the UK and across the EU for a required length of time and could not find any suitable candidates, before they can hire you.
Honestly, sponsored work visas can be very hard to qualify for, and if you don't qualify for the unmarried partner visa, it might be worth considering getting married in the UK and switching to a spousal FLR(M) visa instead (same as an unmarried partner visa, but instead of living together for 2 years, you just have to be married).
Just a note though, that the family visa rules (spouses, unmarried partners etc.) changed in July 2012, and this thread hasn't been updated since October 2011, so much of the advice on it is out of date now and may not be accurate.