The product description of the new sensor stick states:
In our testing, we’ve found that the BME280 sensor requires some burn-in time (at least 20 minutes) and that readings may take a couple of minutes to stabilise after beginning measurements
My own measurements show the opposite. The BME280 measurements are extremely stable and reproducible. I had run a large scale test with nine BME280 comparing them to three DS18B20 under various conditions. Sampling was one measurement every five minutes, power off in between. All measurements are within the limits that the datasheet claims and are near to what the DS18B20 give. No need for a burn-in time or stabilization time.
I think the problem is within the BME280 libraries you use. Your default configuration configures the sensor with filtering and oversampling, but the datasheet (“3.5 Recommended Operation”) states in the section “3.5.1 Weather monitoring” that neither filtering, nor oversampling should be used.
I would suggest re-testing with optimized settings. Of course there are other use-cases for the BME280 than weather monitoring (e.g. as a barometric altitude meter for indoor navigation), but I think most users will do some sort of weather monitoring with the BME280.