You can achieve this in some ways mentioned below:

1. Calculated Column: Create 2 calculated columns, one for each date you want to save. You can look for date formulae here. Based on your text field, you can make your formula add the date to your calculated field.

Note: Calculated columns will not take people & group fields in its formula.

2. Designer Workflow: Create & Run a workflow on item created and updated which checks for your text fields and based on that updates the date fields

3. Javascript: One more way to achieve this is using js on the new and edit forms. Add your logic to js code and add the js in content editor web part on new form and edit form.

Best way would be to use calculated columns.

You can achieve this in many ways as mentioned below:

1. Calculated Column: Create 2 calculated columns, one for each date field you want to save. You can look for calculated date formulae here. Based on your text field, you can make your formula add the date to your calculated date field.

Note: Calculated columns will not take people & group, lookup, ID etc. fields in its formula, Source.

2. Designer Workflow: Create & Run a workflow on item created and changed conditions which will check for your text fields and based on that will update the date fields.

3. Javascript: One more way to achieve this is using javascript on the new and edit forms. Add your logic to javascript code and add it in content editor/script editor web part on new form and edit form.

Best way would be to use calculated columns.