Your summons are pulling aggro because that's literally what they do - they're designed to tank and draw attention. The problem isn't timing, it's that you're expecting them to work like a backup DPS when they're actually your frontline.
Here's the thing: on Adept with a melee build, you don't need to be landing hits while your summon fights. You need to be using that window to close distance safely or set up your own engagement. Summon the Dremora Lord, let it charge in and grab aggro, then sprint around the side and backstab or power attack from behind while the enemy's focused on the summon. You're not waiting for your turn - you're using the summon's aggro as cover.
If you're getting swarmed by adds before your summon even spawns, that's a positioning problem, not a conjuration problem. Don't summon in the middle of a pack. Back up, get some distance, summon behind cover or in a doorway, then engage. Takes maybe 2 seconds and changes everything.
For perks, the one that extends summon duration is worth grabbing eventually since it means fewer recasts mid-fight, but honestly at level 22 you're probably fine just upgrading your melee tree first. Conjuration works with almost any warrior build without needing specific perk synergy - it's just a "oh crap" button or a "this boss is annoying" button.
Don't lean harder into pure melee. This hybrid is working. You're just learning when to summon, which is a positioning thing, not a build thing.