Tip 122 vs. ULN2003

Math looks good.
I think you'll want indiividual logic-level low Rds N-channel MOSFETs then.
Use to connect the - of the strip to Gnd, connect + to +12V.
A part like this for example

Drive from Arduino pin with 150 ohm resistor in series so Gate capacitance does not damage anything (MOSFET internal cap looks like a short to Gnd, resistor limits the current to ~35mA.)