1. I would look to fish a new wire rather than use an adapter. This is what I did.
2. Adapters work based on "power stealing". Ecobee is definitely better at it than Nest (per user reviews), but it can still cause issues especially when none of the systems are on.
3. I would look at the utility company websites to see if they have any deals. They almost always have some rebate offers where I live. e.g. I got my ecobee for < $100 (it typically costs > 200).
Thanks, I do have $100 off rebate for any smart thermostat
1. Do you need one thermostat? Or two? My utility offers a max of two rebates per customer. I got two ecobee's using this for the two zones.
2. I ran wires to both.
I did the thermostat replacement along with the oil to gas conversion. So I had a HVAC tech working with me. I have one of these "Electrical Fish Tape"s:
https://www.amazon.com/SWANLAKE-Fiberglass-Running-Coaxial-Electrical/dp/B07WQ34VF8. (it is more of a "rod" than "tape").
This, coupled with the fishing tapes that almost any HVAC tech will have is a fantastic combination to pull wires through the walls. The electrician pushes the wire from top (using his fish tape/rod) and I stand below and pull the head of the assembly out of the opening using my hook (you can see the hook in the amazon picture) once it shows up through the opening.
Not just for the smart thermostat, I've even pulled Ethernet cables everywhere in my house using this + did one HDMI run to the main living room TV.