xfce4-screenshooter, the default on Xubuntu, has the ability to open it with a program. So download any OCR program in Linux,say tesseract.
Make a script
tesseract $1 | xclip -sel c
save it as scrocr
And choose this program in xfce4-screenshooter, it'll remember it for next time, keeping the drop-down menu short.
This one copies to clipboard, if you want to view it first, then pipe it to `zenity` or `yad` instead of xclip, one of which im pretty sure is pre-installed in ubuntu. And you can copy it out of there normally after checking it.
If you use gnome etc and not xfce, check comparable options in gnome screenshot tool
That bypasses all the copy restrictions.