What shoes required for simple trekking in Georgia?

I'm planning a trip in Georgia, where I plan to visit main cities and spend a few days on trekking in mountains. I don't plan any glacier trekking or climbing, just a bit walking below.

I know the climate is hot, and my feet react badly on heavy shoes when there's hot. I have heavy full leather good, which are quite waterproof (wet snow is killing them eventually, but they resist to wet grass), but in summer I got blisters because of heat and wet. I have also very light Decathlon shoes with their Novadry membrane, which is resistant to shower, but is leaking very quick on wet grass. I've never had shoes with 'real' membrane like Goretex.

I'm not sure, which shoes should I take? Should I buy Goretex shoes and hope they will be lighter for my feet than those full leather ones? Or just take light shoes with light membrane (or without membrane) - and expect there will be not many showers, or that such shoes will dry in a few hours?

I hope that given data enables to answer my question. I hope someone has similar problems (similar reacting feet) and can answer from own experience.