One little trick that's fairly important in FPSs of Marathon
's vintage is double-chording, where you use two strafe keys at once to move about a third faster than normal: As in, point your view about 45° to the left of where you actually want to run, then hold down “forward” and “sidestep right” at the same time to move. The extra speed should allow you to clear many shorter gaps without even grenade hopping, and will make grenade hopping more effective to boot.
Also, lob the grenade a ways ahead of you to minimize damage, fire the assault rifle's gun downward while in the air to add a bit of height at the expense of some length, make sure you're running actually running rather than walking, and if you're really having trouble surviving then look for the 2X shield charger.
This level isn't really very hard, but I bet you're REALLY going to enjoy the next level, Colony Ship.