One drawback is that you can't take system out of rotation as quickly, but this can be somewhat mitigated by setting very low TTL (60 seconds).
If you do go the multiple A record route, do so in combination with one of the load balancers you asked about, and ideally with IP failover. But probably this is a lot more complexity than you want.
There simply is no way to guarantee that a user will remain on one host for the duration of a session, if you use round-robin DNS (or even semi-deterministic DNS like F5 or GSS).
Your best efforts will be thwarted by load-balanced browsing proxies that 'optimise' the user experience by querying primary and secondary DNS, then distributing your HTTP requests across multiple servers. HTH