Today one of my relatives came with 88 corolla, he have 175/70/14 in his 88 with no spacers, neither his car body touches with tyre. I am telling you this just to show you that by installing 14 inch rims, there would have been no problems.
65/14 will reduce less height as compared to 70/13. But in reality I think the problem is with your shock absorbers, check them or go to a mechanic.
My 90 corolla also have some shock problems, as the shocks are not good but they are not too bad.
In my suggestion there is no need to change the rims, save your cost. Just inspect your shcoks by yourself or visit a mechanic.