Blackberry is still a smartphone. Have you tried texting/emails using a swipe keyboard? You might be able to find an inexpensive Android phone and limit the apps to the ones you mentioned. I have a lower model blackberry phone I use as a backup when I travel out of the country, but found I did not get along well with the keyboard versus using a swipe style with Android. Plus, Android let's you use widgets to show the weather, text, and email on the main screen. Just be careful that the specs aren't too low.
Right now I keep a Samsung galaxy j1 mini prime as a second business phone with minimal apps that works fine. It has a replaceable battery and two sim card slots so I can change carriers when traveling or change the battery if the old one konks out. The down side, is that with the smaller screen, I make more mistakes with the tiny onscreen keyboard. If you go for an unlocked phone, make sure that the 3G or 4G radio frequency matches your carrier's.
I will say that my Blackberry goes longer on a charge and has a much better speaker phone than my higher end Android phones though... although maybe the fact that it's 2G and doesn't have a touch screen also saves battery life. Another down side is that my Blackberry doesn't charge with a lot of my universal micro USB chargers.