Getting over 300 was really tough for me too. You'll need something besides strikes / engrams to get you over the edge. The raid is an obvious suggestion. The Nightfall might help a bit too. It's too bad you didn't run SRL for the last 3 weeks to get some high light helmets / class items. But now that that's not an option anymore, I suggest running Iron Banner this week. Once you hit rank 5 you'll get some guaranteed high light items. 300+ at least, probably even a 310+ item. I'm pretty sure it was the last Iron Banner that finally put me over the edge.

It's too late now, but SRL was dropping helmets and class items that were 310-320 once you hit rank 3. Unfortunately, that ended yesterday, but Iron Banner starts later today. Last IB I had some post-match drops that went up to 300, and at rank 3, Lord Saladin gave a package with a 310 rare artifact, and at rank 5 he had another with a 310 ghost (I also had a 316 ghost drop after a match) and a random weapon. I did all three of my characters and two of my wife's, and all the weapons the got from the rank 5 package were around 315. Mine were various primaries, hers both got sniper rifles. I'm assuming this time around it will be the same sort of deal.

Getting gear over 300 is most efficiently achieved by attempting raids. Challenge modes NORMAL guarantee 310 light level weapons. You need a sniper rifle (1000 yard stare is a great choice). Preferably long range primary and a machine gun (you could do with a rocket launcher at the start, but choose a machine gun if you have the option) Once you level up, your engrams will start decrypting at higher level. Court of Oryx tier 3 can also get you some good rewards. Try the recruitment tag for normal raids, destiny Sherpa Reddit if you are just starting and want someone to walk you through, and overall make some friends that need to do the same thing you need to do. Good luck

Gotta run trials, iron banner or oryx raid. Only gonna ever see 300 maxed drops unless you get lucky on a nightfall.