For what you describe, you do not need Adobe Chat for this. You can do this easily yourself. If there are complicating factors Adobe Chat is available.
Computer 1 - Premiere Elements 10
Computer 2 - Premiere Elements 10 (same purchased serial number as Computer 1)
and you want to use Premiere Elements 10 (same purchased serial number as Computer 1 and 2 on Computer 3.
1. Open Premiere Elements 10 on Computer 1 or 2. In the opened project, go to Help Menu/Deactivate and click on Deactivate.
Close out there. At that point, you can leave the deactivated Premiere Elements 10 on Computer 1 or 2, or uninstall it via the typical Control Panel route.
2. Then install your Premiere Elements 10 on Computer 3.
Please let us know if that works for you.
If there are complicating factors, which does not seem to be the case, then you click on the following link, and, when it opens, click on the statement "I still need help" to bring up the Adobe Chat. This type of activation service is offered by Adobe even if it does not involve the current version.
I have pre-set the link for "Serial Number Issues or Activating My Product".
As I understood your issue, this should be a straightforward matter which you can handle.
Please let us know the outcome.