Hi Mark,
I’ll start with saying that you should consider terminating your engagement with this lawyer.
Now with regards to my own experience, I started my process with the property search by going through the aggregator websites and then contacting the real estate agents for the shortlisted properties.
Obviously the real estate agents were representing the interests of the seller, so once I have chosen the best property only then I hired a lawyer to do the conveyancing. And again I found them myself independently of the real estate agent.
Thinking of conflict of interests, I had a very minor one later on in the process where the real estate agent acted as an interpreter for me during the signing of the deed (which is a legal requirement in Portugal if any party of the transaction do not speak Portuguese). I could see no risk in that as by that time a certain trust was already built and also I was able to read and understand the deeds in Portuguese myself anyway.
Hope this helps. There is also an excellent post here by xraydelta covering the property research part of the process.
By the way, if it can be of any use to you, DM me for the real estate contact as I worked with a big name agency in Portugal, so I am sure they offer properties in low-density parts.