If Electrum stops working, I can just use Electrum offline to build the wallet and sign the transaction. Then I can broadcast it with another wallet. Am I correct?
Absolutely. Provided you have a working copy of Electrum saved somewhere, or can source a copy from someone else online (which will almost certainly be possible outside of the situation of a global catastrophe), you can restore you multi-sig wallet offline, move the address to an online computer and use it to create an unsigned transaction, move that transaction back to your offline wallet to be signed, and then move the signed transaction back to your online computer to be broadcast. You can use a service such as
https://www.coinb.in/#newTransaction to create your transaction for signing, and you can use the broadcast function on the same site or pretty much any block explorer.
Also, shouldn't I be able to build the same wallet in another software wallet that allows multisig for BTC?
That was my question above. Electrum seed phrases are not the same as BIP39 seed phrases. Almost all wallets use BIP39, and so you would not be able to just import your Electrum seed phrases in to another wallet. You could extract the extended private and public keys from your Electrum seed phrases and import
those in to another wallet, but I'm not sure which other wallets support generating multi-sig wallets from imported keys.