It looks as if it characteristic of the signal rather than a problem with the scope. Even a digital scope would have a similar issue.

One way around this type of problem I use when developing MCU software is to exploit an unused GPIO on the MCU to act as a trigger for the scope.

Connect this additional GPIO to the other channel on the scope (or the external trigger if all channels are in use).

Within your code toggle this GPIO at the time you wish to trigger the scope.

As Transistor indicates in another answer the issue is probably caused by jitter in the signal - even that might be helped by issuing the trigger from within your code as as in some situations you may be able to arrange that the trigger has the same jitter as the signal you wish to observe and so cancels out. For example a GPIO trigger just before the character is sent out the UART should reduce the jitter dramatically rather than triggering from the previous character.