Grumpy_Mike:
but I'm guessing it has something to do with the additional load affecting the LM35's output.
No.
Measure the voltages on the analogue inputs and see if you can see the change with a meter.
Write a much simpler sketch that just reads the two channels and prints them out and see if you still have the same problem.A schematic would be good. What is that blue component close to the temperature sensor.
Hi Mike,
It's worth mentioning at this point (as if you haven't worked it out already), that I have an extremely rudimentary understanding of electronics, but a 25 year Microsoft based (VB6, ASP, C# etc) programming background. I did an electronics course (Basic Certificate of Electronics) about 20 years ago, but most of it's a fading memory.....V=IR is kinda familiar 
Re: A schematic (and to MarkT)
My elementary electronic knowledge does not allow me to draw one by hand, let alone in a format I can post here.
I tried the schematic view in Fritzing, but it's not the view I'd expect to see if someone asks for a schematic, but please let me know if you want to see it anyway.
The blue component is a 100nF cap across the power inputs of the LM35, apparently acting as a 'decoupling capacitor'. The idea as far as I can understand is to make the LM35 draw power from this cap when it needs to do it's reading, in an attempt to ensure a more stable voltage at the Vcc leg of the LM35 when the reading is taken. Please feel free to point out any errors in my understanding of this cap - it's all pieced together from reading stuff on the web, along with posts on this forum, so it's quite possible I've got it all wrong 
I have run a 'test' version of that sketch that has nothing but 2 readings from 2 components and got the same results as soon as the second component was introduced.
Would you mind telling me 'in dummy's guide format' (as my knowledge level obviously requires) the tests I need to perform to measure the voltages to get the voltages you suggested?
Many thanks in advance