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Hello John,
As a general rule of thumb, a thorough check of the Low Voltage power supplies, both for voltage on spec and ripple are both important and yet, an easy check.
Not just because problems at the power supplies may cause, at times, unpredictable problems that are hard to make sense of (even odd ones such as everything working BUT syncing), but also because finding an off-spec power supply that has not yet caused a serious damage, can save you from watching the inevitable magic smoke, and the need for a much more complicated repair.
Having made sure the power supplies are healthy, the next step (in your case, where the scope seems to be working to a great extent) is to get acquainted with the operation of your scope, to be sure that the position of the controls are all correct for the operation that you're attempting and for the signals you're applied.

More specific to the symptom you mentioned, here are the basics...
1. Exercise all selector levers, push-buttons and pots a dozen or more times... to make sure that you cleared any false contact that may be present, due to oxidation of contacts, or dried lubricant crud.
2. Make sure Horizontal Display mode ganged push buttons is set to A sweep (button "A Lock Knobs" pressed),
3. That the A and B Time/Div dials are locked (that both the clear dial skirt and the gray knob are aligned and moving together) and a suitable sweep speed is selected.
4. Trigger mode ganged push buttons is set to AUTO (button AUTO pressed).
5. That the A Trigger level pot is set to middle range.
6. Vertical signal level has some healthy 4 or 5 graticules in height (and that you use only one channel (for now)
7. A trigger coupling lever is set to AC
8. A trigger source lever is set to NORM.

From this starting point, if it doesn't sync right away, then there must be some issue at play... and it will become a matter of narrowing it down, by fiddling with the controls, one at a time, in a combinatory fashion... For instance:
A. From the starting point above, try to move (slowly) the A trigger level pot, from CCW to CW. If no avail, put if back to center.
B. Try to change the Source lever, from NORM, to the specific channel where you have your vertical signal fed to (CH1 or CH2), and repeat sweeping the A trig level from full CCW to CW.
C. If still no avail, change the Triggering coupling from AC, to HF REJ, to LF REJ and DC... for each possible coupling, try the steps A and B above.
D. If neither work, use a BNC "tee" and a coax cable, and try to feed the same vertical signal to the A EXT input, in combination with changing the the Source lever to "EXT" or "EXT÷10". At this step, try playing with all combinations of the Coupling lever while sweeping the A trigger level pot from CCW to CW.
E. If neither work, then you can still try to feed a vertical signal from the AC mains (or the output of an AC transformer), select an appropriate sweep speed and change the Source lever to "LINE". While at this step, exercise again the Trigger coupling lever and the A trig level potentiometer...

As an extra resource of information, while at each of the lettered steps above, you can try to press the trig view switch.
Under normal conditions, on a working oscilloscope... while pressing the TRIG VIEW push button, you should see on the screen (instead of your Vertical signal), a "fac simile" of the triggering signal that is coming out from the A trigger amplifier, past all the coupling, source and level controls....It should **normally** display much larger on the screen than your original vertical signal is showing.
While you press the TRIG VIEW button, rotating the A TRIG Level potentiometer should make the signal cross the screen from top to bottom (while turning the A trig level from CCW to CW).
If you see the trigger signal while pressing the TRIG VIEW button, and if it reacts to the A trig level control, and the scope still doesn't sync, then you have a problem on the triggering circuitry past the triggering pre-amplifiers.
If you don't see the trigger signal while pressing the TRIG VIEW button, or it stays stuck at either the top or the bottom of the screen, then you have a problem on the trigger input circuitry or the trigger pre-amplifier.

Hope this helps you move further...

Rgrds,

Fabio