Mazal tov on the birth of your baby!
1. A bedikah performed on the morning of the day of a planned hefsek taharah can count as a hefsek taharah after the fact, as in your case. So you may rely on the bedikah you did this morning.
2. Using a paper towel is not advisable for a hefsek taharah. Soft cloth is preferred. Paper towels can be rough. They can also break up, and sometimes might impede feeling that one reaches all the internal crevices.
In a pinch, a pair of white underwear can be used as a bedikah cloth.
If a woman omits her moch dachuk, her hefsek taharah typically remains valid. Because the area is sensitive, we usually advise against performing a moch dachuk in the first cycle postpartum.
We hope this helps and that you’re able to get bedikah cloths for the rest of your clean days.
Please let us know if we can be of further assistance.