Use Google Voice/Google Hangouts. They give you a free local number that you can use to call and text people while you are online and logged into Google. If you use Google Chrome, you can even install an app for Google Voice/Hangouts on your computer that keeps you logged in (unless you need to update Google Chrome).
I personally love using my Google Voice/Hangouts. Unlike you, I do have a phone but it's a prepaid one and I typically don't like using it because I have to pay for every text and call. But with Google Voice/Hangouts, I can text and call people for free so it's just more convenient that way and I only use my cell phone just in case.
If you have any further questions on how to set it up, feel free to comment and I'll get back to you as soon as I can! I hope this helps!