I modified the circuit as you said and it fixed the random lighting.
As it should.
This set up works to run one strip just fine, but when I try to add in the other strip the program seems to stop and start over in the same point every time, about 3/4 of the way through the first strip.
So we now need to see your code, posted in "code" tags according to the instructions. First thing to check is that you have set the correct number of LEDs.
Oh, wait, you say it does work with one strip? Right! Next two points:
What exactly is the reason for providing power every 30 LEDs? That would be rather difficult with these strips being covered.
So they all get five Volts. The less they get, the less reliable they are and less able to show correct colours.
Would being powered at each end do a similar task? Each strip is 240 LEDs long.
That would be the absolute minimum. It's also why they have the red and white wires at both ends. If - when you get it working - it fails to show white away from the power when all white, then you may need to modify it further.
240 LEDs? 14.4 A on full white. You may think you never need full white, but if it gets sent at some point, even briefly, and the power supply cannot provide that, well what do you will imagine will happen?
Link for LEDs - https://www.amazon.com/gp/product/B00K7UHPEC/ref=ppx_yo_dt_b_asin_title_o02_s00
can work on USB power
Picture for New Setup