You should be able to change armour, except I just thought that maybe you have built armour above the level of your champion, in which case you won't be able to equip until you have reached the required level.
You do not need to dismantle equipment unless you need materials, but you say you have lots of them.
Each piece has a level as mentioned above. If your Champion isn't to that level yet you won't be able to equip the Champion with it. As a rule I kept my last piece before dismantling. All others I dismantled at first. Towards the end I had 2 pieces of the various equipment per slot. JUst to play with mostly. Almost exclusively the higher the equipment level the better. NOTE: Each piece has different bonuses/amount of bonuses so make sure you use the piece that will help you the most. NOTE 2 : The equipment you can buy for Runic Coins are pretty much needed for the last levels. Its possible to get by without them but they GREATLY enhance your chances of clearing floors. I was stuck on my main account for months on the last floor and rooms before I broke down and spent the 5mil Coins for the .... sword I think it was. Breezed through all remaining on my first try after that. Something to keep in mind. Yes, 5mil Coins are alot but after months and months of grinding Hel it was a relief to finally finish. Well worth the Coins.