Voltage Regulator Not Providing Enough Current/Voltage
The chip die will get hot almost instantly and may go into limit before the heat conducts out to the heatsink tab.

Power dissipated in the device is \$ (V_{IN}-V_{OUT})I = (14-9)\cdot 1.1 = 5.5~W\$.

I don't see any decoupling capacitors on the regulator input and output. You may have a high frequency oscillation going on.


  Even with a heat sink, can this regulator handle the the stress we're asking from it?


It can if we can keep the inner chip temperature low enough.



Figure 1. Thermal information from TI datasheet section 6.4.

I've highlighted the two relevant figures here. With no heatsink the chip temperature rise will be 22.9°C/W so you could expect a 125°C temperature rise which puts it just about on the 150°C max.



Figure 2. TO220 heatsink.

This one has a thermal resistance of 13.4°C/W. If we get good thermal conductivity between the tab and the heatsink (use thermal paste) the total resistance will be the series sum of all the thermal resistances = 13.4 + 1 = 14.4°C/W. At 5.5 W the temperature rise will now be 80°C. This is a drop of 45°C and will be a big improvement. Bigger heatsinks will reduce this further.