Airport check-in agents (as opposed to the airline website, or unmanned check-in kiosks at the airport) are, in my experience, usually happy to assign seats together at no charge if they're available (though of course you won't get to choose which seats those are). Just approach the check-in desk together. However, on crowded flights, depending on the airline, there may not be any adjacent seats left by the time airport check-in opens.
Checking in separately online at the same moment, if anything, makes it even less likely that you'll just happen to end up together. The systems I'm familiar with don't just go 1A-1F, 2A-2F, etc.; window and aisle seats are distributed before middle seats.
As Willeke mentioned, it's possible to swap seats (though on some flights I've taken, the crew has requested people to stay in their original seats until after takeoff). For parents and children travelling together, the crew will do their best to get them together (for example, they'll offer free drinks to someone willing to switch to a middle seat to accommodate it), but for just a couple of adults, you're likely to be on your own. (Note, however, that you are also empowered to buy a drink for someone to entice them to swap.)