Altaholic here… I’ve been away for a long while, came back right after tww started, playing some new, some old toons.
It is REALLY exasperating that some of them start with a full set of 20+ slot bags, but others start with one good old 10 slot red bag. 
All bags seem to be soulbound, which is like the FIRST thing they should’ve made warbound.
I just had to vent, even if others have vented before me, it’s been annoying ever since I restarted playing.
Buy wildercloth bags on the AH. Fairly cheap and each one is 32 slots. If thats too much gold, then netherweave bags cost next to nothing and will serve until you can afford the better ones.
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I still have 12 Traveller’s Backpacks. 16 slots, match the main bag, and are not soulbound. So as I level alts, I buy bigger ones later and toss these in the WB for a future alt.
You can get a 34 slot from Dragonflight in the first zone. Once you hit 70 cuz leveling is rather fast, you can also run out and get 2 more. One from Ringing Deeps, and one from Hallowfall. Should help a bit.
I usually just buy 32 slot bags as soon as I log in or find the one server where they are the cheapest and buy them to stuff in the warbank. One server was selling them for 75g and I just bought enough to fill the bank then logged in and snatched four for every character.
I can feel your pain. I myself am very particular about which bags I use
Bags are archaic game design anyway. WoW needs to switch to a weight/encumbrance system immediately for maximum immersion.
Add 2 more bag slots to each toon. OR… upgrade our backpack to the # of slots we have in other bags.
Or maybe a quest… to upgrade the # of slots in backpack. But can’t do quest until all other bags are 36.
My backpack is 20 slots… I think that’s including the extra slots for having authenticator.
Backpack should have most slots, maybe 50 
I’ve got a warbank tab half full of 30/32 slot bags to toss onto new alts.