Hallo Raj,

you don't need to apologize. I only wanted to focuse the point, that recommendations of datasheets should strongly be followed. Often datasheet does not explain all mechanisms of internal circuitry in detail, but instead recommend some additional components, like this input filter. Only experience tells then, that those additional components are a must and are not optional.

As Michael already told in detail, your signal routing is too high impedant. If you cannot decrease source impedance drastically, you should insert a buffer for each source before signal is reaching CD4051. Buffer should sit as close as possible next to source, while signal routing between each buffer and CD4051 can be extended over a bigger area, especially when shielding is used.
Behind CD4051 locate this input filter (100kOhm + 100nF), followed by ICL7135. Keep distance between components of input filter and ICL7135 as short as possible!

This should be your basis for experiments. Also, use very low multiplexing speed at the beginning of experimenting, when multiplexing your 8 channels. I highly assume, that with a rate of 1 channel per second, your application will work quite well. I further assume, that you will not observe any unexpected error.

Now, that you are sure that everything is working fine, you can start to play a bit with mulitplexing speed and components of input filter. But not earlier!!!
Increase multiplexing speed and decrease time constant of input filter up to the point, that you observe some unacceptable error. Keep in mind not to decrease R to less than about some hundreds of Ohms (best: R >= 1kOhm). Smaller values can make trouble, means, outputs of buffers can become instable and begin to oscillate due to high capacitive load. Also, output current of buffers can increase drastically, when C must be charged or discharged!

By doing this testing and adapting you get a feeling for your circuit. May be you find out, that by carefully matching all parameters you can find a very good compromise. But may also be, that you will realize, that with this circuitry you are not able to achieve adequate multiplexing speed. Then you must try an other approach.

Good luck,

Kai