Both, usually switching between pushing and pulling and alternatively both pulling and pushing at once.
The third option available for inputting arm power is, by lying extra-flat on the seat back and gripping the brifter hoods,
my arms are straight: In this position, my shoulders can do the push-or-pull/ push-and-pull thing.
My shoulders can do this much longer than my wimpy arms can.
As you found out, your legs easily overwork your arms.
Another way, my favourite method, to put down all the power you're capable of generating, without stressing your arms,
is to rock your bike in time with your pedal strokes.
You're riding a D.F. bike, reconfigured as a recumbent bike.
Realize this... it's literally true.
Now, rather than read my rambling description of how I fling my bike around,
go to Youtube and watch videos of professional sprinters racing for the line.
Watch how they handle their bikes.
Learn to do what they learned to do.
Good luck,
hope this helps and
you're welcome!