Not necessarily full power, but there is a peak efficiency point of a fuel cell usually quite a way of it's notional peak power. It's better to let the fuel cell sit at that point and just churn out x watts constantly than trying to vary it up and down to meet the exact demand - the instantaneous response (and storage function) of a battery is far better suited for that. The auxiliary power to start a fuel cell comes from the battery, and it's also worth noting that there is a point where the power the fuel cell generates becomes so low that you're better to turn it off (the fuel cell requires a certain amount of power to operate it's 'balance of plant' - compressors, condensers, cooling etc)
As for 'do you have to run it at full power all the time' - that depends on the fuel cell you use and the duty cycle you're running it on. The ideal would be to size the battery and fuel cell such that you could run it constantly at a low, but efficient power so that you never quite run out of battery at the worst point of a duty cycle (eg just climbed a gradient from stationary and are about to brake to a stop again downhill). There's no reason why you couldn't turn it on and off as necessary when the battery nears full, but the most efficient system is one that isn't so big that you carry around extra weight/cooling requirements for it to sit off half the time, nor one that's so small you have to run it harder than peak efficiency and loose range (or leaves you stranded without enough power!).
Also don't forget that as the fuel cell's only operating emission is water, there's no reason to turn it on and off as you go through stations or urban areas!