TPS63070 Buck-boost not bucking or boosting correctly
It appears you have the correct resistors in place for the feedback circuit. The only thing that appears out of place is the 330nH inductor. Looks like they have a 1.5uH in the data sheet. It should still technically regulate to the voltage you need though..

Have you soldered this part by hand? I've had issues in the past where I've soldered a part incorrectly and then it fails to regulate. (either by damage or bad connections) Always good to check on a development board if you can. TI's are dirt cheap and worth every penny.

Also, is there anything connected to that rail that could a source of back powering? It appears you said you have a power source at about 5.8V but if you have any other power sources in the system I would definitely take a look at that too.