I am also a new cyclist (4 months or so, ~800 miles) so take my advice with a grain of salt, but I have seen a lot of improvement on hills over that time-frame. There are a few things that I have focused on while climbing that I think have helped me.

First is, learn to use the full pedal stroke on hills; 360 degrees of power from both legs.

Second, sit up to expand your lungs and take long deep breaths. I have found taking long deep breaths helps me conserve energy and allows me to get the oxygen to my legs that I need. Sitting up with your hands on the tops or the hoods will expand your lungs a bit and allow you to breath easier. No need to be aero when riding up a hill.

Third, do more hill repeats and use those repeats to experiment. I start my repeats in the lowest gear and spin up the hill. Then next time up the hill, I increase one gear. I continue increasing the gear I use on each subsequent repeat. In addition to training both spinning and strength, I think this exercise has really helped me understand my gearing better. It has taught me what level or resistance gets me up the hill fastest, and what level of resistance gets me up the hill with the amount of energy exerted.

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On a side note, huge congrats on the weight loss! I am on a similar journey myself, also 6'2. I am down to 190 from 245 with a goal of 180. When I started cycling I was at about 230 so loosing all that weight is probably the biggest factor in me getting better at hills.