Per this Google Cloud document, by default, when an organization is created, and to minimize disruptions, all members of your domain are granted project creators and billing account creators, but not for the organization itself, which would be visible but not modifiable. They will only be able to modify the organization when the G Suite or Cloud Identity super administrator has set the correct permissions.
I understand that you have tried this before, but I recommend to re-verify that the modification is being done by a G Suite super admin. For further step by step guide to assigning admin roles to users you may check this support page. If the super admin was not able to assign the admin role to the user using the guide, please privately email me your organization ID, along with the email of your super user for a quick inspection.
On the other hand, as a member of the domain, (as explained above) you are a project creator (by default), and if you are an owner or editor of the existing project, you should be able to migrate the existing project to the organization without being an organization administrator, for further details you may check this help article.