Accuracy is not as important as precision though. For accuracy, you can usually just calibrate it out against a known reference (you usually need to do this anyway to get consistent results across multiple sensors). The only case when you can't is nonlinear accuracy, which is rare but I've seen it with the DHT22 actually.

Precision (stability of measurements) is what you really care about. That said, the BMP180 isn't so great at that either. The temperature sensor is I think a secondary function used to improve the pressure estimate. It doesn't give very stable temperature estimates in my experience.

If you want a direct replacement for the DHT22 (and I did, god I hate that module for a variety of reasons) the HTU21D-F is great. The BME280 is also a big improvement over the BMP180. If you just care about temperature, the DS18B20 is the way to go on performance and price, while if you really want high accuracy and precision then the MCP9808 is the way to go. I use the MCP9808 as my reference to calibrate against, with a mixture of the other three in various applications.