Yes. All cards fail in the new system but work in the old system.
Yes. I could. However, I'm unable to isolate my module to run those sketches.
However, I've managed to find the problem. Sometimes, the problem exists where you least expect. After re-soldering all the connections, rechecking for continuity and scoping the signals, I was totally thrown off to realize that SD card was getting the correct signals but still unable to read it. I tried to add a beefier 3.3V supply but no luck. I had tried to even load the exact same sketch that works on older system but it will not work.
Finally, I loaded the exact compiled hex code that is present on the older system and it worked! So, it had to be my code. But all I added was couple of debug statements to see why it wasn't working. After much investigation, it turned out to be a mix of two problems.
When things stopped working in the beginning, it seems that for some reason Arduino 1.6.5 that I have wasn't building properly. It had to be quit and restarted for the build to produce the correct output.
To debug that, I had added a debug statement like:
ms.print("ABC"); //this is where the problem is
As strange as it may seem, it turns out that trying to open a file after printing a STRING with MinimumSerial doesn't work. Note that if I print a single character at a time, it all works but a double quoted string is an issue. I've verified it over and over.
I would be delighted to know why that is happening.
Anyways, thanks sterretje for encouraging me.