Indeed it is strange. Both that it only happens on certain sites, and the Mac is assigned 35 rather than 1. Makes me wonder what device is 1.1. But that would be the router itself if it is self-assigned as 192.168.1.1.
Connecting wirelessly rather than wired doesn't change anything on your end. The router is still going to use 192.168.1.35 to communicate with the Mac.
I would try to reset things to the defaults. Shut down the Mac, then the router. Reset the router to factory defaults. Typically, this is done by holding in a small, recessed button for 5 seconds or so. Check the manual for your router to see how to reset it.
After the reset, turn the router on and wait for it to come to a full ready state. Then turn the Mac on. They will negotiate a new IP address if necessary.
Try the sites again.