Seeming only changes how things look
You omitted an important part of the spell's text:
The changes wrought by this spell fail to hold up to physical inspection. For example, if you use this spell to add a hat to a creature's outfit, objects pass through the hat, and anyone who touches it would feel nothing or would feel the creature's head and hair. If you use this spell to appear thinner than you are, the hand of someone who reaches out to touch you would bump into you while it was seemingly still in midair.
A creature can use its action to inspect a target and make an Intelligence (Investigation) check against your spell save DC. If it succeeds, it becomes aware that the target is disguised.
You can entirely disguise that you have weapons
Since you can use it to make things that are present non-visible (e.g. when appearing thinner), you can make weapons entirely disappear in the illusion, for example, to appear as an unarmed peasant, as long as you adhere to the restrictions, i.e. you cannot make the creatures overall appear larger or smaller or have a different body type than the spell allows.
You only can become aware of the disguise, not "see through" it
If combat breaks out, and the disguised or invisible weapons cause damage, then the victim will realize something is wrong, because the weapons interact with them physically.
Will that lead to the illusion automatically failing like for minor illusion, or is all they can achieve to be aware something is wrong? Minor Illusion has the same language about not holding up to physical interaction but states:
If a creature discerns the illusion for what it is, the illusion becomes faint to the creature.
Here, the spell does not say it becomes faint if you succeed on the Intelligence (Investigation) check, only that the creature "becomes aware that the target is disguised". This would mean that even if you are aware of the disguise, it does not become faint, and you still cannot look through it. Seeming is a much higher level spell, and ususally, higher level spells have more powerful effects.
I think you automatically become aware as you get hit, because that is a kind of "physical inspection". You only need to spend an action if you do not physically interact, but suspect something is wrong.
Interacting with disguised weapons
When they hit you, you'll know the disguised creatures are hitting you with something invisible or different from what it appears as.
What are the mechanical effects of fighting against an invisible or disguised weapon?
If the entire attacker is invisible, they get Advantage on attacks. Here only the weapon is invisible, so the invisible condition for the attacker does not apply. And the effect of not being able to see your attacker and what they do at all should be larger than the effect of just not being able to see their weapon. As the spell does not say it gives advantage on disguised weapon attacks, rules-as-written it would not give Advantage to attacks.
Intuitively however, it should still be harder to dodge or parry an invisible weapon, or one that has other dimensions than the one you see. I think it would be up to the DM to give some circumstantial bonus here, or not, depending on the situation.