There are two places where you must manually add these dependencies yourself.
The first is in the ModulePackage.targets file. In here add a line in two places:
<Copy SourceFiles="$(MSBuildDnnBinPath)\$(AssemblyName).dll" DestinationFolder="$(MSBuildProjectDirectory)\Package\bin"/>
<Copy SourceFiles="@(SqlDataProviderFiles)" DestinationFolder="$(MSBuildProjectDirectory)\Package\%(RecursiveDir)" />
<Copy SourceFiles="@(PackageManifestFiles)" DestinationFolder="$(MSBuildProjectDirectory)\Package" />
<Copy SourceFiles="@(PackageTxtFiles)" DestinationFolder="$(MSBuildProjectDirectory)\Package" />
<Copy SourceFiles="$(MSBuildDnnBinPath)\WebGrease.dll" DestinationFolder="$(MSBuildProjectDirectory)\Package\bin"/>
Edit the last name to reflect the name of the dependency your project needs.
Secondly, edit your .dnn file. Near the bottom of the file you need to list your assembly like so:
<component type="Assembly">
<assemblies>
<assembly>
<name>ContactUs.dll</name>
<path>bin</path>
</assembly>
</assemblies>
</component>
Without this DNN won't know to copy your assemblies to the bin folder.
As a note, I've had issues with using different versions of Newtonsoft. At least in DNN 8 and earlier the admin menu depends on a specific version and if you replace it you'll encounter a lot of issues.