You could buy some better or more cushioned insoles for the boots, but they would probably still not be as soft as trainers
Have a look at trail shoes and fell running/off road running shoes. There are so many, just Search t'Net to see the sort of things available
It would also be a good idea to have a browse round a large outdoor shop, such as Go Outdoors
Or have a look at Pete Bland Sports, who specialise in fell running & orienteering
If you go for lighter, lower cut shoes or trainers it is inevitable that you will get wet feet more than in boots - that's the trade-off for lightness & comfort. Many people accept that, adopting the attitude of splashing on, but being light & comfortable even though you're wet. This type of footwear will dry out more quickly, as you wear it
You can get trail shoes with waterproof linings; their disadvantage is the higher price, and that they are still low cut, so much more likely for water to go over the top; and when that happens they will take longer to dry