Warning: I have no experience of the HP1345A/3562A, however your display reminds me of the Digital Equipment Corp (DEC) GT40 vector display, on which I spent many hours landing the lunar module (a.k.a. moonlander).
Your photo appears to me that your are missing some ‘Z’ blanking as there are far too many lines visible (Z appears to be permanently stuck ‘ON’).
The ‘Intensity Control’ U25 appears to be a good place to start. Is the O/P on U25 switching or is it stuck?
Furthermore, you mention "intensity started flashing" which could also indicate a problem with the Z circuit.