How might I triumph when it's all said and done my last name transformed me and my significant other are lawfully isolated I kept my birth name however from that point forward

How might I triumph when it's all said and done my last name transformed me and my significant other are lawfully isolated I kept my birth name however from that point forward we have taken on his grandson and he would like for us all to have similar name we are attempting to accommodate I observed all of this I understand or discover that the town hall lost the marriage permit and didn't record them and I didn't take my better half's last name at the time since we're guarantee I made my father when he was passing on how might I right this and triumph ultimately my last name changed legitimately without causing a great deal of distress all the while and doesn't make any difference with Social Security that it was not at any point change since I kept my family name and all of my father how might I do that also and we are legitimately isolated