There's a lot of layers here that could be influencing what is perceived from different points of view, so...
Yes, it is entirely possible that changing the DNS server that you use for resolution changes your IP address on the receiving end.
The responses you get for IP addresses from Content Delivery Network (CDN)-based services can change because folks use Global Server Load Balancing (GSLB) tricks to route you to the closest web server in their "cloud" based on the IP address that asked their DNS for the website name. As an example, if you switch from your local ISP to use OpenDNS, the IP address that makes the final leg of the DNS resolution request changes, so it's entirely possible that the CDN GSLB DNS servers perceive your client to be in a different place geographically than when your ISP DNS server does that final resolution leg. This then means your web browser connects to a different IP address to open the HTTP connection, and that different network destination might route your IP address through a different Network Address Translator (NAT) box so now your IP address is NAT'd to a different public IP address and that's what the web server sees when HTTP transport is connected.
This is all very common these day.
If your address falls in 10/8, 172.16/12 or 196.168/16, then you must be NAT'd and this all applies.
If you have a static public IP address, then this may still be happening (I've seen it, and scratched my head since it doesn't make sense), but is much less likely.