What EXACTLY are the documents needed to submit to probate court to obtain letter testamentary?

Your lawyer should submit a death certificate, the original Will and an application to probate a Will.  This is not a DIY project. Since 1843 Texas has allowed estates to be settled independent of court administration (other than admitting the Will to probate, appointing the administrator, reviewing and approving an inventory and recording creditor's claims), this may well be the extent to which Texas probate law is "simplified."  Fortunately, that makes it much cheaper and faster than in other states.