The resistors aren't current limiting for the LEDs. To be honest, I'm not quite sure what they do, but connecting the TLC output to the ULN input didn't work without one. Putting a 10K one from the ULN input to Vcc as suggested in that thread does work, although the PWM is reversed for some reason, that's not important. (I can easily fix it in software).
I'm using the TLC as a way to get more outputs from the arduino. With it I have 16+ PWM outputs, without it I only have 6. The current limiting bit isn't needed, because as you pointed out, the ribbon has resistors built in.
Are there better devices than the TLC and ULN to use, or is this the best combination?
Check out shiftpwm for pwm on shiftregisters, but it seems to me, you need 185ma per channel, and you need 30 channels, so think you will need some very stout shift registers.
You could use 30 transistors, but I dont know if it would be better.
I cant see your strip, but im pretty sure that those resistors are to limit the current at 12v, which is why they are a 12v strip. Its possible that you have something different than im imagining though.
The TLC and the darlington arrays are both sink devices, they both control the the ground (-) side which is why you are getting the opposite of what you expected to get. (you are sending low to the darlington expecting it to turn on, they work the opposite way, which is why you are getting the opposite results in your PWM). I think if you put an pnp transistor between the TLC and the darlington, it might behave the way you want, but damn thats ugly, and the ULN wouldnt be very useful.
Most of the strip setup that I've seen use mosfets, with 10 strips, it sounds like you will want to use 30 transistors of some type (Im guessing pnp or p-channel)and your tlc.
You can do PWM with shift registers, checkout shiftpwm, its pretty cool, but I think that maybe too much current, since you want to run 185ma on all 30 channels (10 RGB channels). each output on the shift register will need to be able to handle 185ma, and each 8 bit shift register will need to be able to handle 8 times that (1480ma, or about 1.5A)