Happy to report that the fix described by dantera
(post 65 on page 5) worked for me as well.
Fixing the burnt out board trace was not enough, but after I replaced the transistor next to the burn, everything was back to normal.
The transistor on my board was an SMD (surface-mount device) with label 3F. After googling, I found that this corresponds to BC857, which is a PNP general purpose transistor.
The local electronics store didn't have BC857, but they did have BC856, which is close enough. Note that this transistor doesn't directly pass the current for the bulbs, but simply helps control the relays. Therefore, it doesn't have to support extremely high voltages (BC857 is rated at 45V, and BC856 is 65V).
Recommended way is, of course, to find the exact replacement part that was originally on your board (just google SMD and whatever code is on the transistor). If you don't get the exact same transistor, make sure you get the same type (PNP in my case).
Word of caution; if you don't have any soldering experience, I wouldn't recommend tackling this. The transistor and board contacts are tiny, and you are more likely to destroy the board than to fix it if you're not careful.
Personally, this was my first time soldering an SMD component, but I have lots of experience with soldering ordinary components.
My capacitors were fine, but make sure to check yours in case one or more need replacing.
Hope this info helps somebody.