Is what the questioner does—placing a cloth after urinating for an hour, then performing istinja (washing), and when drops of istinja water come out, performing istijmar (drying with a stone or similar)—correct? And what is the ruling on his previous prayers?

If what you feel is merely a whisper (waswasah), then pay no attention to it, turn away from it, and do not investigate what you imagine to be coming out after urinating. Build upon the original principle, which is that nothing has come out. If you were to accustom yourself to sprinkling your trousers after urinating, and then apply what you feel to this practice, this matter would cease for you. However, if you are certain that drops are coming out, then wait until their emission stops, then perform istinja' and pray. There is no harm in using stones (istijmar) for these drops if that cuts off the waswasah, and istijmar is permissible even with the availability of water, provided that its requirements are observed.