Is it permissible to rinse the mouth and gargle with water to alleviate dryness of the mouth and throat during the Ramadan fast, and does that affect the validity of the fast if the fasting person feels something has been swallowed after expelling the water?

Your fast is valid and is not nullified merely by doubt about swallowing some water during rinsing the mouth and gargling, because the default is the validity of the fast. Swallowing saliva or moisture from the mouth after rinsing and spitting out the water does not invalidate the fast. So, if some wetness remains after rinsing and you swallow it with your spittle, it does not break the fast. The remnants of mouthwash water with saliva after expelling it do not harm, nor does swallowing saliva after rinsing, even if it was possible to expel it. However, gargling should not be done, as it is similar to excessive sniffing (istinjāq) which is forbidden for a fasting person for fear that water might reach the throat. What remains in the mouth after expelling water is considered saliva and does not take the ruling of water, and therefore does not break the fast.