Inappropriate readings of pulse and temperature sensor (LM35)
The output stage of the LM35 is famous for being a little moody… in my experience it often tends to oscillate, if the signal line is longer than 5-10cm (typical if externally mounted). Also it is somewhat weak, as you already noted.
Look with a scope at the output signal, if you see a sawtooth the LM35 is oscillating because of the excessive capacitive load (the output wire is seen as a capacitance). It may be eventually break, you did well trying another one.
There are tricks to make if work slightly better but the real solution is to use another kind of sensor with better drive capability or buffer it (a simple voltage follower is enough) before entering the long wire.