If you're at someone else's house and you don't have access to towels or a hair dryer, here are some steps you can take to try and dry your damp shirt:
Wring out the shirt: If your shirt is soaked but not dripping, you can try gently wringing out the excess water. Be careful not to stretch or damage the fabric while doing this.
Blot with absorbent materials: Look around the house for any absorbent materials you can use, such as paper towels, tissues, or napkins. Press these materials against the wet areas of your shirt to soak up some of the moisture.
Use a fan or open window: If the weather is warm and dry, you can try standing in front of an open window or using a fan to help speed up the drying process. The air circulation can help evaporate the moisture from the shirt.
Roll the shirt in a towel: While you mentioned you can't use towels, you can still try rolling your shirt in a dry towel to help absorb the moisture. Gently press down on the rolled-up towel to encourage absorption.
Change your environment: If there's a room with better ventilation or air circulation, consider moving there to help your shirt dry faster.
Patience: Unfortunately, air-drying takes time, especially if you don't have access to towels or dryers. Be patient and allow the shirt to dry naturally over time.
If it's a dire situation and you really need a dry shirt, you can explain the situation to your host and ask if they have any spare clothes you could borrow temporarily. Most people would be understanding and willing to help in such situations. However, if that's not an option, just do your best with the available resources and make yourself as comfortable as possible until the shirt dries.