There is a way to move servers, and to download your archive, but I’m not sure whether it moves your old toots to the new server (probably not?). I haven’t used it myself yet. Here’s a page about it though: https://nerdschalk.com/how-to-switch-servers-on-mastodon-everything-you-need-to-know/
As for finding servers, here’s a list of resources for browsing lists of servers.
https://joinmastodon.org/servers - a sort of flagship list of instances (a.k.a. servers), including many big ones
https://fediverse.party/en/portal/servers/ - more general list of servers organized around particular topic themes
https://joinfediverse.wiki/Local_instances - instances centered around a particular geographic area
https://joinfediverse.wiki/Special_interest_instances - instances centered around particular interests
https://joinfediverse.com - smaller servers that are ready for new users
https://mastodon.help/instances - a big list with some filtering options
https://instances.social/ - another big list with some filtering options
https://fediscience.org/server-list.html - science-related servers
https://www.fediversesearch.com - big filterable list (can filter for Mastodon instances)
Lesser-known servers probably won’t be there but you can look around on other people’s addresses to see what servers they’re using and check out those domain names yourself to see if they’re servers willing to accept new folks.