Can I ask what they suggested and what you ended up with eventually?
The second doctor told me to rip the insoles and replace them with some orthopedic, custom-made ones (my insurance covers a pair or two a year) and also prescribed me some knee pads with metal rods on the sides that I must wear whenever I run. He also sent me to physiotherapy for a month where I learned exercises to strengthen the knee and core muscles. Whenever my knees start to hurt (typically after not exercising for a while) these exercises make it go away.
Sounds like pes planus aka flat foot. Yes, run of the mill Adidas would do nothing, you need a shoe with a good side support and custom insoles.