Beavers in the Moscow region are definitely a problem that a lot of dacha owners run into. You're right that it's best to contact your local forestry office or the regional wildlife protection department right away - they handle trapping and relocation legally. Usually this is done in spring or fall, when the animals are more active. It's pretty much impossible to drive beavers away on your own, they're super stubborn and once they've taken a liking to a place, they won't just leave.
Until the specialists arrive, you can try some temporary measures - fence off valuable trees with metal mesh about a meter high, since beavers rarely gnaw above that. You can set up noise deterrents or use capsaicin-based repellents, but the effect is temporary. Some dacha owners block the entrances to their burrows (carefully, so you don't hurt the animal), but again that's not really a solution, just a delay.
Get in touch with the forestry office in your specific district in the Moscow region - they should keep records of these cases and know how to handle things legally. When beavers are trapped, they really do get relocated to other places, usually somewhere that's been arranged. It takes a while and requires patience, but it's the only truly humane way out.