How can I check everything is fine with a cold wallet (offline Electrum)?
If you have exported the addresses on a txt file (or the master public key as Charles-Tim mentioned), then you can check their balance. You can either paste your addresses on block explorers and check it via the browser, or import them as a watch-only wallet (on electrum). You can also import them on a Bitcoin client, such as Bitcoin Core. The balance will appear after syncing. Note that the last option is the most preferable if you want to be independent of any block explorers/electrum servers.
I checked the addresses from the wallet on the blockchain
On the blockchain.com on generally on the block chain?