To transfer android phone text messages to computer, you can try Coolmuster Android Backup Assistant, it's easy to use, have a try.
I have using AirDroid for file transfer for a while and love about it. The messages on my phone(Galaxy S6 Edge) almost reaches the limits. so i need to back up them on my computer, then i can delete the old ones. However, i didn't find such a function on AirDroid, am i wrong? Is there anyway to get that? It's urgent, please help if you knew about it. thanksIf you can't do that with AirDroid, you can search some android backup tools on google.