Dropbox disabled their old API. I added support for the new Dropbox v2 API in Textastic 6.3.2 in April last year.
You are still using Textastic 5.3.1.
Textastic 6 is the first paid upgrade since version 1.0 was released in 2010.
Implementing split view multitasking, making the app universal and supporting iPad Pro required a rewrite of large parts of the app. Charging for Textastic 6 allows me to continue to dedicate resources to improving it.
You can purchase Textastic 6 on the App Store: https://itunes.apple.com/us/app/id1049254261?ct=support&at=11lNQP&pt=15967&mt=8
Customer support service by UserEcho