[SignCheck] Avoid holding exclusive writes on log files

See https://github.com/dotnet/source-build/issues/5041

SignCheck currently holds exclusive writes on its log files, so when SignCheck times out, the following errors can occur when trying to access the log files:

```
The process cannot access the file 'D:\a\_work\1\a\SigningValidation\signcheck.log' because it is being used by another process.
[WARNING] Asset D:\a\_work\1\a\SigningValidation\signcheck.error.log failed to get process

The process cannot access the file 'D:\a\_work\1\a\SigningValidation\signcheck.error.log' because it is being used by another process.

[WARNING] Asset D:\a\_work\1\a\SigningValidation\signcheck.xml failed to get process
The process cannot access the file 'D:\a\_work\1\a\SigningValidation\signcheck.xml' because it is being used by another process
```