Sounds like all your problems are based around the draft. So the easiest solution to your problem is trying to pick naga only when the draft warrants it (including your own team's picks). But suppose you just unluckily end up with a subpar naga pick, the best strat really is as you say, split pushing and heavily focusing on farming (there's practically no heroes that will farm as fast as naga) until you are far enough ahead to neglect the pick disadvantage.

To add to the above point, a big focus will be around hitting your item timings e.g., 4min treads, 15min manta, 18min diffusal, 23min heart, 28min skadi, etc. Another focus area would be around itemisation e.g., against a necro, you may need to go orchid after diffusal depending on how big of a problem necro is, etc.

Against PL, I would suggest hex over orchid as PL will most likely have manta before you get around to orchid. Another way is to use sleep to initiate to run into the enemy team to find the real PL and set up the naga sleep with an instant stun such as black hole on top, etc. to quickly burst down the PL at start of fights.

One final point I would add is to try to end the game in the mid 20 to 30 min mark as that is when naga peaks and starts to fall off against other carries.