Hi @jmashkovskiy
From what I see - the root of the issue will be the check box ‘Run at Beginning’
When using the ‘Run at Beginning’ feature, this occurs before the data has been compiled for any <<tags>> . If you uncheck this, It will update the information on the record after the successful Document Generation and the dynamic tags will populate.
The only other gotcha around Dates and insert updates may be the format Salesforce wants data to be in. Similar to this article: ( feel free to try the tags you already have and adjust to similar tags in the article if needed)
Hope that helps and should get you to the next step
Brent