If the thermal paste is over 6 years old it is probably losing a lot of its efficiency by drying out.
I would first make sure that the GPU fan is not dirty or running slower than normal under heavy loads. That the GPU Heat sink fins are clean and not obstructed or filmed over with gunk preventing efficient cooling. That you have good air circulation inside your PC.
Otherwise I would probably replace the Thermal Paste with a high quality Thermal paste and pads if those are damaged when replacing the Thermal paste.
This website that gives out Maximum Operating Temperatures for various CPUs and GPUs mentions the R9 380 Maximum Operating Temperature is 94c. But not sure how reliable this Tech site is:
If the above data is accurate then you don't have to worry if the temperature reaches 70c under heavy loads. It is still way below it Maximum Operating Temperature of 94c.