You might be able to do both. Depends on the school I think. For example, I am going to get the GEM stipend and my university rep informed me I might be able to find a paid research position separate from GEM if I wanted. You should ask your rep the details of your fellowship, some require RA/TA and some don't. If the university just gives you a stipend then also ask how you can get a RA/TA position for additional money.
Me personally I would take the for sure stipend, and try to find a position at the university for more money. I think that's better than just the "chance to apply to positions" in my opinion because that doesn't seem like a sure thing.