When you say you go between different phones, do you mean that you have multiple phones each on their own cellular plan, or do you mean that you swap a SIM card into different devices regularly?
If the former, are you using the same Gmail account for all of the devices?
Google Voice should be able to do this if you can use one Gmail account. Google Voice besides being capable of replacing your voicemail and being a super tool for visual VM and handling calls also has texting. These texts are visible anywhere you can sign-in to Google. So if you had multiple devices where you used GV for texting all of the texts would be there when signed in. The same is true for VM and call logs that GV handles.
There are other ways, recently I saw a post where the user was getting messages on their phone and a tab. Unfortunately I don't know how they did that but I found this article speaking of another app based solution...
https://www.google.com/url?sa=t&rct...ngle-number/&usg=AOvVaw2l3f9VxMH4GTrZIvOjlUvp
Google and YouTube also have solutions.