I am not familiar with that particular temperature sensor, but the circuit looks fine and should be stable. However most temperature sensors do require about one minute to stabalize, Before you reset it.
Also, do you have any 1 uF Tantalums as Bypass Caps on the 5 volt supplies, going to ground? That would be a Good Idea.
According to the datasheet, the LM35 takes up to 4 minutes to stabilize in still air.
I don't know why you have all those op amps. You don't have to use all of the op amps in the package just because they are there. More op amps will cause more drift and error. You also don't need R1 and R2. They have their own drift, and they cause you to have to raise the gain, which also causes more drift and error.
With a 5 volt supply, the LM324 is only guaranteed to reach 3.3v maximum on the output . Is 33 degrees an adequate high limit for your circuit? If not, choose an op amp whose input range includes GND (as does the LM324), and whose output range is rail-to-rail.
You also don't need R3 and C1 unless you are driving a large capacitive load as would be the case if your LM35 is remotely located (which I realize may be the case).
I've posted a simplified schematic below. Note the 100nF decoupling caps added to the ICs' power pins. This helps prevent oscillations that might otherwise occur. I've also reduced your gain control range, which will make the circuit much less sensitive to drift in the potentiometer.
lm35dz circuit - pls comment
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