Scuol would probably offer the best choice of accommodation and is the transport hub for the Lower Engadin. Other villages to consider would be those directly on the railway line, such as Ardez, Lavin and Zernez, but options would be more limited in these villages.
The railway station in Scuol (Scuol-Tarasp) is on the edge of town, so if you decide to look for accommodation in Scuol, given your plans, it would be more convenient to choose one of the hotels near the railway station rather than in the centre of town.
If you are willing to consider the youth hostel, it is adjacent to the station. You could get a private double room with ensuite and breakfast buffet for CHF 61.80 per person per night. Non-members pay a daily surcharge of CHF 7. It is modern, clean and comfortable, but basic, as you would expect of a hostel. I have attached some photos I took when I stayed there in early October last year. I would happily stay there again.
Although your main plan is to hike, I would strongly recommend you find time to include a visit to the tiny village of Guarda, and to walk through the old part of Scuol.
If you are interested, I have written trip reports about Scuol and Guarda:
Wet weather alternatives could include the National Park Centre at Zernez and the Monastery of Son Jon at Müstair. Both can be reached with the 811 bus from Zernez station, which is about half an hour by train from Scuol. A PostBus trip over the Flüela Pass might also be an option.
>> Ultimately, we will make our way to Chur or St Moritz to catch the Bernina Express onward to Milan for our departure.
There are hourly trains from Scuol-Tarasp to St Moritz, requiring a change in Samedan (journey time 1 hour 24 mins).
There are also hourly trains from Scuol-Tarasp to Chur, either direct or with a change in Landquart. Journey time is 1.75 to 2.0 hours.
Both routes are scenic.