Fallout – Can you finish the game siding with Caesar as a woman

No.

There is a clearly delineated "Point of No Return", whereby setting out upon the final quest requires you to explicitly tell an NPC that you'd like to do so, from which point you can no longer explore the Wasteland, and once completed, the game has an ending, from which point, you can't continue playing. While this point is fairly obviously telegraphed, if you happen to not see it coming, the final quest involves returning to the Hoover Dam, after completing quests involving President Kimball.

You should be able to salvage this situation.

My suggestion? Get some backup. From Nipton, continue on to Novac ASAP and recruit Boone to watch your back - he'll go a long way to helping you survive Legion Assassins.

If you're still getting roasted every time they manage to catch you, make sure you're fighting smart: Melee weapons are better than guns for overcoming DT in the early game, and armor penetrating ammo, while not common, should be available a few rounds at a time from most vendors. Stick to the roads and watch your back, and follow the main quest as single mindedly as you can. Once you get to the Strip and meet Mr. House, you'll be able to clean your slate with the Legion and go from there.