I don’t know what your circumstances are, your skill level or anything else besides that you played Lyonar. Ergo, I’ll make the assumption that you play Lyonar for the most part and that your collection doesn’t have access to higher-cost pieces.
Countering Swarm Abyss is about proactive management of the swarm. While the Abyssian player may spawn a lot of minions, they are for the most part very weak, and you should take the effort to kill them off - even with your general - unless you are going for the kill that turn.
If you have to kill a large number of small minions, you have access to Tempest as Lyonar, but you also have access to Blistering Skorn.
[details= Blistering Skorn]
3 mana 3/4
Opening Gambit: Deal 1 damage to all minions and Generals.[/details]
Blistering Skorn not only kills 1-hp minions, but also takes off durability of almost any artifact. You can also could consider playing Bloodtear Alchemist
[details= Bloodtear Alchemist]
1 mana 2/1
Opening Gambit: Deal 1 damage to an enemy general or minion[/details]
Bloodtear Alchemist not only kills off minions with 1 hp, but also assists in destroying artifacts. It also can be a target of buffs spells or abilities at a very competitive price point.
Between your attacks, Tempest, Blistering Skorn and Bloodtear Alchemist you should be able to deal with the swarm. On the other hand, you still have to deal with it’s supporting elements which are generally a little more durable.
Enter Sun Bloom
[details= Sun Bloom]
2 mana Spell
Dispel a 2x2 area of the board[/details]
Dispel effects don’t get rid of minions, but it does remove the text. This is important - decks such as swarm Abyssian are a great big machine of moving parts that manufacture your defeat. However, they are finely balanced machines - remove one piece and the machine grinds to a halt. Support minions such as Bloodmoon Priestess, Shadowdancer and Shadow Watcher are all lose value when they don’t have their text - they are instead understatted minions. And as Lyonar, your minions are all at least on curve or far above the power curve compared to their counterparts.
[details=Bloodmoon Priestess]>4 mana 3/3
Deathwatch: Spawn a 1/1 wraithling nearby this minion.[/details]
[details=Shadowdancer]>5 mana 4/4
Deathwatch: Deal 1 damage to the enemy general and heal your general for 1 health.[/details]
[details=Shadow Watcher]>3 mana 2/2
Deathwatch: Gain +2/+2.[/details]
Now onto the topic of Abyssian card draw. This is possibly the hardest part of any card game - the management of your resources. Mana is a resource. Cards are a resource. Life is a resource (except for the last one - you need that!). Just because they can afford to fling spells at you does not mean that you should as well. If you need card draw, cards such as Blaze Hound are fine.
[details=Blaze Hound]>3 mana 4/3
Opening Gambit: Both players draw a card.[/details]
Between your overstatted minions and the damage from your spells and abilities you can usually put pressure on your opponent and beat them before they can set up their plan.
I apologize if this post comes off as dismissive or condescending of your opinions. However, Swarm is very beatable. All decks are beatable, you just have to understand the game. I hope this helps.
Though a well-piloted or lucky Songhai will always be frustrating