If you double-click on a text with the selection tool, the text tool should get selected and the text-cursor is set into the text.
What do you want to do with the text object? Do you want to edit the text (then double clicking is OK) or do you want to push the text around the artboard, then don't double click.
Sounds to me as if you somehow have turned off selection preview. Pressing Command-H toggles between viewing selections and hiding selections. If this is the problem, pressing Command-H should fix things.