I noticed that moving windows of TortoiseProc.exe between monitors with different DPIs causes blurry text.
In TortoiseProc.vcxproj I see EnableDPIAwareness is set to true while TortoiseMerge has PerMonitorHighDpiAware set. So you should also use PerMonitorHighDpiAware for TortoiseProc.exe as moving window of TortoiseMerge is good. So this different DPI setting should fix it.
Btw, TortoiseGitProc.exe has the same issue