What should I do wit my signifigant others car after his death and I am not on the loan.

No, you can't trade it in unless it was left to you in your significant other's will, but that will would still have to be probated. Assuming the car was titled in your significant other's name alone, his/her interest in it would transfer as directed in his/her will or, if there was no will, by the laws of intestacy.  If there was a will, it must be probated before your significant other's property can be disposed of.  If there was no will, his/her property would generally pass to his/her closest living relative or relatives, not to you, although these laws vary depending on the state where the decedent lived.  However,  there may need to be a court proceeding to administer the estate before any property can be distributed.