Avignon has a population of about 92,000 people and the closer you get the the Centre Ville the more dense the population becomes. It is not at all a place that one would find greenery or a countryside feel. If you won't rent a car and stay in the countryside then you are just not going to get that countryside feel anywhere in Avignon. I don't mean to burst your bubble but I do mean to give you a dose of reality so that you'll be prepared for what to expect.
I agree with FM above. Most hotels are located at the central area where you will find "country side" greenery. Naturally, the less dense parts of the city have far fewer hotel choices. Also, they are most likely at least 30 minutes on foot from the city center, occasionally via roads that aren't great for walking.
If you cannot afford La Mirande then maybe try something in Villeneuve-lès-Avignon just across the Rhône. I do not recommend the walk from the Best Western du Lavarin to the city centre which takes you through the worst district of the city. You could stay there and take a bus, but even that hotel is not countrylike. It is next door to the crematorium.
It is a contradiction in terms. If you do not have a car you need to stay close to a train station. Train stations are in the city centres. And it is very hard next to impossible to find a hotel in a city centre that feels like countryside.
There are, of course, hotels in the outskirts of cities, like the one you have found in Saint Remy, but they require longer (and not always pleasant) walks or taking taxis or maybe taking buses (if they are close to a bus stop which is not always the case).
Since you appear to be not on a small budget, you might be ready to spend 30 or 40 Euros per day for taxi rides. In this case, you may look for countryside hotels around Arles which is a smaller and even more agreeable town than Avignon. Tripadvisor map search is excellent.
I also like the Avignon atmosphere more than Arles. It's just a personal opinion though. At Arles, I stayed just outside the center (I had a car). At Avignon, I stayed centrally almost next to Palais de Papes. On subsequent trips, again with a car, I stayed in smaller towns/villages instead.
As to Villeneuve-lès-Avignon, that was the area that came to mind when I first saw your question. However, you have a lot less accommodation choices.