I have the same cheap modules, and also had similar problems with them.
So, I discovered that they do not take 3-12V for Tx and 5V for Rx, like most of tutorials are saying, but are completely different modules.
These modules you have are new versions, and if you read their specs, Tx must be fed with 2-3.6V and Rx with 2.2-5V. If this does not help, try changing the pins, for me non-pwm digital pins work better.