multibit IS online and synched - so no idea why export doesn't work.
Sometimes it has to be offline to export the keys. I don't know why, that's just my experience of how it can work.
I wouldn't consider upgrading until you make a backup, and I would advise against upgrading to the latest multibit classic. The guy who started the linked thread waited for three days for his multibit classic transaction to confirm. Eventually he had to pay a miner to get it confirmed.
https://bitcointalk.org/index.php?topic=1909433.0This is the transaction.
https://blockchain.info/tx/026ca669edad3ab32320b22688694f89b873bc068e4cbc8cd7ce73d2daf552c5Although the network was more overloaded in early May the number of unconfirmed transactions could suddenly increase and leave a multibit classic transaction stuck in limbo.
The latest version of multibit classic has a slider which you can use to increase the fee. This quote explains it, but the maximum it will let you pay is 0.0005 BTC per KB which isn't enough at times when the network's extremely overloaded.
The latest multibit classic (0.5.19) has a fee slider in the preferences. It's maximum possible fee is 0.0005 BTC per KB, and there are times when that's not high enough to get a fast confirmation. This is a screenshot of it.

For a transaction sending one input to one receiving address and a change address paying a fee of 0.0005 BTC per KB would result in you paying a fee of about 0.0001 BTC for a transaction, which is about 28 cents. That's nine times lower than the fee this website recommends to get fast confirmations.
https://bitcoinfees.21.co/If you switch to electrum and use its dynamic fees feature your transactions won't get stuck in limbo.
If you still want to risk upgrading to the latest version of classic there's a guide here.
https://multibit.org/help/v0.5/help_upgrading.html