A single word for "moved around a lot"?

Solution 1:

How about "nomadic" or "nomads", as in "We were very nomadic" or "we were nomads."

Nomadic:

ADJECTIVE

Living the life of a nomad; wandering.

Nomad:

NOUN

1 A member of a people that travels from place to place to find fresh pasture for its animals and has no permanent home.

‘the withering of their grasslands forced the nomads of the Sahara to descend into the Nile valley’

https://en.oxforddictionaries.com/definition/nomadic https://en.oxforddictionaries.com/definition/nomad

Another option might be "transient" is the word you're looking for? As in, "we were very transient."

Transient:

NOUN

1 A person who is staying or working in a place for a short time only.

https://en.oxforddictionaries.com/definition/transient

Edit: moved Nomad/Nomadic to the top; left Transient in so the comments still make sense.

Solution 2:

We were peripatetic; Oxford Dictionaries:

Travelling from place to place, in particular working or based in various places for relatively short periods.

‘the peripatetic nature of military life’

Solution 3:

Relocated works well:

"I was in a military family. By the time I was nine, I lived in 6 different cities. We relocated frequently."

Relocated (Oxford)

Move to a new place and establish one's home or business there.

There's an example of the word being used in the context you have mentioned on The U.S. Department of Housing and Urban Development website:

In 2012, there were over 1.2 million children in U.S. military families, and more than three-fourths lived in households headed by enlisted military service members. Military families with children frequently relocate, often moving across state lines or to foreign countries, and move every two to three years, on average.

Solution 4:

Though not a verb, I would restructure the sentence to use something like "rootless" or "unrooted"/ "uprooted."

"A rootless, though relatively happy, stage of my life."

http://www.dictionary.com/browse/rootless

rootless
[root-lis, roo t-] 

adjective
1.
having no roots.
2.
having no basis of stability; unsteady:
a rootless feeling resulting from economic and social change.
3.
having no place or position in society; not in accord with the environment:
the homeless, rootless wanderer.