Surely the whole point of texts is that they don't interrupt like a phone call and you can respond when you please?

On the other hand 5000 seems like an awful lot - there can be too much of a good thing. However I am an older person and its seems younger people send 60 texts per day so perhaps 5000 is not out of the way: CHART OF THE DAY: Number Of Texts Sent - Business Insider

Perhaps you can pre-empt annoying them by saying, 'let me know if I am interrupting something or if you are busy'. Then they can politely let you know if this is the case.