starcraft-2
I am a story person, and I played StarCraft 2 because of it's story.
Now that I've finished all of the StarCraft 2 campaign parts (Wings of Liberty, Heart of the Swarm and Legacy of the Void) is there anything that I can do besides multiplayer battles (fighting another person or an AI)?
If I'm understanding your question properly, in order to have two Heart of the Swarm licenses you need two Wings of Liberty licenses.
Since Heart of the Swarm is not a stand alone expansion, you need to have all the prerequisite games in order to play it (each copy).
You can see for example, on this Cataclysm box art for World of Warcraft, that it requires "Wrath of the Lich King to play". Wotlk says it requires TBC, and TBC requires the base game. It is expansions all the way down. This trend stands true for all Blizzard games to date, as they don't release stand alone expansions.
In the starcraft 2 lore panels at blizzcon 2010 and 2011 they answered these questions. The canonical choices are the A choices, so
Raynor chose Tosh over Nova
Raynor chose Dr. Hanson over Selendis
Raynor chose the Tunnels in Belly of the Beast, not the platforms in Shatter the Sky
Best I can find for the tunnels answer is the entire lore panel from 2011.
Best Answer
Other than playing multiplayer, there are a few different options you have:
1. Achievement Hunting
All three main campaigns feature achievements rewards for various... well, achievements. Earning Achievements can also unlock character profile pictures, medallions and Battle.NET profile levels.
They can be broken into a few categories
1.1 Mission Completion
Every mission has 3 achievements to earn. You can replay missions at any time to earn their achievements.
Unless otherwise specified in the achievement description, achievements can be earned on any difficulty. As you complete mission achievements, you'll be working towards...
1.2 Story Arc Completion
Each campaign is broken into a series of 'chapters' that are comprised of various related missions. Completing all achievements on a series of missions will earn you a 'Mastery' achievement for that section.
For Wings of Liberty, Mastery is based around certain character's missions, such as Tosh (Covert Missions), Tychus (Artifact Missions) or Matt Horner (Dominion Rebellion) etc. In Heart of the Swarm and Legacy of the Void, missions are grouped by planet.
1.3 Campaign Completion
Awarded for completing the campaign with certain goals in mind. For example:
1.4 Challenge Achievements
From the Wings of Liberty campaign selection, there is an option to run through some practice challenges. I'll expand a bit on this later
1.5 Multiplayer / Vs. AI
You mentioned briefly that you want to avoid multiplayer, but there are achievements for playing multiplayer and vs AI as well.
1.6 Other
There are a tonne of other achievements - for example, ones for maxing out different upgrades on Hyperion, utilising Kerrigan's and the Spear of Adun's powers etc etc. Take some time to browse through and look for the interesting ones you want to go for.
2. Fanmade Maps and Campaigns
There are some fun fan-made campaigns out there, there are some in the in-game 'Arcade option, otherwise have a look at SC2 Mapster for more campaign-oriented maps, or Curse for multiplayer maps.
Of course, simply searching for 'Starcraft 2 Maps' or 'Starcraft 2 campaigns' will return more than a few results as well.
2.1 Starcraft: Mass Recall
One of the more popular campaigns out there, Starcraft: Mass Recall is a dedicated fan remake of the original Starcraft and Starcraft: Brood War storylines in Starcraft 2.
Check it out here
3. Challenges (Wings of Liberty)
Alongside the Wings of Liberty Campaign there are a series of Challenge missions - giving you a limited unit composition for you to survive against multiple waves of enemies. You must decide which units are used against different waves.
As well as bragging rights, completing these missions will earn you more precious, precious achievements.
4. Prologue and Epilogue (Legacy of the Void)
There is a series of missions set before/after the events of the main Legacy of the Void campaign.
The Prologue is made up of the prerelease missions, and detail more of Zeratul's journey to uncover the prophecy.
The Epilogue wraps up what happened to Narud/Duran, the main characters, and the Void.
5. Hidden mission (Wings of Liberty)
There is a Hidden mission in Wings of Liberty. To find and play it, see: How do I get to the secret mission in Wings of Liberty?
6 Co-op Missions (formerly 'Allied Commanders')
Team up with a friend for some co-op fun in the Co-op missions introduced in Legacy of the Void. You take on the roles of various commanders from the StarCraft universe, and battle through a series of special scenarios together with a friend, utilising your unique abilities, upgrades and bonuses to overcome different scenarios.
New abilities and units are unlocked as one progresses through the mode, along with gaining experience.
You can read more about it on the Wiki
7 Make your own Campaigns & Maps
Starcraft 2 has a Map Editor, just like Starcraft 1. You can use it to build your own custom maps and campaigns. Give it a whirl!
Nova: Covert Ops (Coming SoonTM)
Blizzard has added an option to pre-purchase a new campaign known as 'Nova: Covert Ops', most likely centered around the titular character.
Whilst not available yet, it's worth keeping an eye on. You can read more on the Wiki