Can I get the IP address of class A as that is the only preferable IP than the class C IP.

Well, this is true that the Class C networks may appear less desirable than class A networks. But the problem is that the network address of class A consists of only 8 bits, only 126 class A networks are possible. All such addresses, have long been assigned, so although you might prefer a class A address, you are not likely to receive one.

I have read somewhere that unfortunately, class C addresses are also becoming scarce, even they are 24-bit network address which can yield 16 million possible network addresses.

A new TCP/IP standard, IPv6 will provide 128-bit IP addresses.