I have a 2004 CRV that began to have a shimmmy when I replaced the OEM tires. The rims were bent and replaced with after market alluminum rims. The shimmy continued to occur at highway speeds [70+ mph]. I have my tires rotated and balanced every 5,ooo miles. The shimmy is not constant and will come and go while driving on a straight road. It is annoying. Would replacing the tires help. I have always had the scheduled maintenance done on the CRV. I have about 104,000 miles on it.
If the rims were bent, someone hit a curb or had an accident, probably throwing off the front alingment. You may have damaged front suspension components that need replacing, or a front end alignment at the very least. Check the alignment and front suspension components first.