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.

Among other features, Textastic 6.1 added Google Drive remote connections. Textastic 6.2 added new fonts and the ability to open Git repositories from Working Copy as external folders. Emmet support and a JavaScript console was added in Textastic 6.3. Textastic 6.4 adds support for iOS 11.

You can purchase Textastic 6 on the App Store: https://itunes.apple.com/us/app/id1049254261?ct=support&at=11lNQP&pt=15967&mt=8