Raster cell size changing with new projection

First, as @Vince said in the comments you want to be using Project Raster rather than Define Projection. Project Raster lets you switch between known projections, where as Define Projection is when your data does not have a defined projection.

Second, take a look at the Environment Settings, specifically Cell Size. The setting allows you to force the output pixels to be a certain size, so you could set this to be 0.5 meters. Note, that any kind of reprojection or warping of rasters is going to adjust the pixels slightly, in terms of size or shape (which is why you initially got 0.35m pixels). When you force them to remain 0.5m resampling of your data is going to occur. Depending on how you are using the data this may or may not be an issue.

In this GIS SE post about raster values changing after reprojecting take a look at @SonOfABeach's answer for some additional details on how reprojecting effects pixel values.