How to stop a switched current affecting an IC?
Try adding a couple of capacitors across Res3 and R4, e.g. 100nF, to slow down the switching and minimise EMI. Also add some bulk capacitance to the touch sensor power rail (e.g. >100uF electrolytic) and the +7.5V rail. 
Let us know the results.  

Also, you can try reducing the sensitivity of the touch sensor. Make sure you ahve followed all the guidelines in the datasheet regarding PCB layout and noise immunity - notice it is recommended to use a dedicated voltage regulator if the power supply is shared with another electronic system:





NSI50350AST3G and PWM

The mention of the capacitors above and in the datasheet is to reduce the higher frequencies that can causes issues when switching large currents at high speeds. Note that "high speed" here refers mainly to the rise time of the PWM signal rather than it's primary frequency.
If we leave the capacitors out, and we switch using the Arduino pin the rise time at the base will be fast (which means the transistor switching will be fast) I don't know about the ATmega specifically, but most modern microcontrollers have very fast I/Os. Many can switch in under 10ns, take for example this scope shot I just grabbed from a dsPIC:

 

You can see the rise time is just over 5ns, this is pretty fast. I'll assume it will be faster than the ATmega, but we'll assume the ATmega switches in 10ns. Remembering that any waveform can be made up from a sum of sinewaves (see Fourier Transform) a good rule of thumb to calculate the significant spectral power density (very roughly - the frequency components large enough to be worth bothering about) from the rise time is:

Fknee = 0.5 / risetime

Fknee is the point where the spectrum rolls off much quicker (e.g. like the -3dB point of a low pass filter) so this is a significant point.  

So for 10ns:

Fknee = 0.5 / 10e-9 = 200Mhz(!)

Now the transistor will switch a lot slower than this, the datasheet gives a value of 100ns maximum for 30V, 750mA and a base current of 15mA. According to a simulation model I found, it is indeed pretty quick. So 100ns still equates to 20MHz, faster than we would like. Luckily, our base current is not 15mA, so the rise time is more like 600ns, which gives us a knee frequency of 1.2MHz, getting much better. Let's have a look at this:



And the simulation with rise time shown (under "Diff cursor2 - cursor1"):



The current risetime is only 985ns (open in new tab for larger version)  

With capacitor added

Okay, now let's add the capacitor and see what we get:



And the same simulation we now have a ~15.6us current risetime:



This is fine for an LED PWM, since the frequency will be low (e.g. 100-300Hz or so) and significantly reduces the high frequency component.  

BUT  

The (rather large) downside is that the transistor base current is not sufficient to switch the 350mA fully (notce the maximum current in the plots above, and hence now disspiates more heat, since it spends more time in between fuly on/off - to be precise, it doesn't turn on fully)
This not be efficient and you don't have your LEDs as bright as you want them (if you do the sim, it's disspating around 0.25W)   

SO

Replace the 4.7k resistors with 1k resistors, and then you should have enough base current to switch the full 350mA. But hang on - what about the rise time if we change the resistors? it decreases to around 2.5us (not so bad), but you can increase the capacitor to 220nF or 360nF if you wish to keep the risetime slow.