This Seeking Tech guide will help you defeat Shinryu without using dragon lances.
Preparation
Before the battle, make sure you learn the Flare, Curaga, Esuna and Arise magic spells. I also recommend that you master all of the jobs, and then use Freelancer for all party members.
In terms of abilities, three of your party members should focus on healing and protection. The fourth should use Spellblade with Rapid Fire.
When it comes to equipment, make sure you equip Ribbons if you have any. You should also give your most powerful weapons to the main attacker. It is also very important that you equip Coral Rings, since these will absorb Tidal Wave and block Atomic Ray.
You can buy Coral Rings in the secret armor shop that can be found in the Phantom Village location.
Battle
As for the actual battle, start out by using Mighty Guard if you have it. If you don’t then you can use the Shell and Protect spells instead. Next, you should have your attacker use Spellblade, and then select Flare.
For the rest of the fight, the attacker should use Rapid Fire. The other three party members should be on standby and should focus on healing immediately after the enemy attacks.