The type of resampling surely affects the nature of the values, especially the largest and smallest ones. The fraction of negative values in your data is minimal, so I wonder if it’s a problem at all.
Reprojection would lead to further resampling, thus chaning your data even more. You can check if different resampling methods lead to other results. As Marco said, if the values are negative from the beginning, then it’s not a matter of processing.