The steps in the above procedure will work fine on your version but you may be missing a step or two.
You should still be able to get to Disk Utility via Launchpad, however. It should be listed under the Utilities or Other folder in Launchpad.
Locate the "Disk Utility" app and open it
It should look similar to this:
In the left hand sidebar, choose your external Hard Drive volume.
On the right hand side of the page, click "Verify Disk" (If the procedure uncovers errors, please let me know for further instructions to repair)
Next, click "Repair Disk Permissions"
Once the process has completed, restart your Mac and try to move a file to the external drive.
If it does not work, in the Finder, click on the Go menu and choose Computer.
Locate your external drive and Control-click on it, then choose Get Info from the pop-up menu.
Does it still list your acces as read only? If so, click the orange padlock and authenticate with your admin accound password.
Then change the Read Only option next to your user name to Read & Write.
Try to move a file to the external drive.