You really can't flush tampons. They snag on old tree roots that inveigle themselves into cracks in old pipes and at seams in pipes, and somewhere along the line there will be a blockage. You might as well flush fabric nappies or cotton handkerchiefs. Yes, they will disappear from sight, but someone, somewhere will end up fishing them out of the sewers. Allegedly 'flushable wipes' are not flushable either. If any of those items make it to an open body of water you will find them all over some beach in the zone by the high tide mark. The reason is they are not designed to disintegrate.
Tampons are not made of cotton wool. They are composed of cotton, rayon (made from wood pulp), 'polyester, polyethylene, polypropylene, and fiber finishes. Aside from the cotton and fiber finishes, these materials are not bio-degradable. Organic cotton tampons are biodegradable, but must be composted to ensure they break down in a reasonable amount of time' (Wiki) And yes, I know that's just Wikipedia, but I think it is safe to disregard whatever Tampax and OB, etc. say about flushing them.
OP -- Wrap your soaked tampon heavily in TP and put it in a ziploc baggie. It will go safely in your bag until you find a bin outdoors to dispose of it if you can't find one in the house. I know people who have undersink cabinets with their bathroom bins in them. So maybe have a rummage around for a bathroom bin next time you need to dispose of a used tampon. If you already do this, then disregard.
I keep a bathroom bin in a little cupboard on which I have stencilled 'Bin' just to keep people from flushing pads or tampons in desperation. It's tacky but heyho. It means people can dispose of anything they want to whenever they need to and they don't have to waste time looking for it.
I have teenage DDs and a DS and all their friends traipse through my home frequently. I find it unreasonable to expect that none of them would ever need to dispose of menstrual products, hence the bin, and it is small and emptied a few times a week. I use a liner in it. I would be mortified if any of them felt any discomfort about a tampon or pad they had to dispose of.
*Zoe is surely about 15 and hairy of hand?