I am using the web design form (not Survey123 Connect) and I have two questions.
How do I change this text: "Press here to submit another survey."?
Currently we haven't support customizing "Press here to submit another survey." I will suggest you to contact Esri Support. Our Support team will assign an official enhancement number for your records. Similar requests from other customers can then be attached to the same enhancement request, which helps us assess demand for the enhancement and prioritize it accordingly.
And if you prefer to open the survey in the same parent target, you might want to use js api to achieve this.
Hope this helps,