I had the same troubles, though on another aircraft type. Everything went well in sim, except for some of the last sim sessions, my OEI approaches were like I was flying an ILS for the first time.
Localizer and flight director went like windshield wipers!
For my part, I found what caused my trouble:
1. I wasn't enough after the speed. Being in close to the speed-unstable region, thrust must be managed properly. I didn't (in time), so with an engine out, large thurst changes became necessary.
2. The changing thrust affected the slip/skid, so rudder needed more attention, whilst trying to get the speed back to where it should be.
3. The out of trim condition caused the heading to wander, as well as flying in a crab meant the aircraft went in another direction than heading was pointing - causing flight director readings to be unusable, until becoming in trim again. For me it was the first time with a crosspointer FD - before I was used to the "fly-into" type, so I had flight director needles and LLZ/GS crossing one another ... and instructor would NOT let me fly raw data.
4. By this time, localizer would be all over the place, and soon all the parameters of speed, thrust, localizer, glide slope, heading and maybe more would be out of control, resulting in a slalom ILS.
The solution was simple - it starts at 1. above:
Keep a good eye on the thrust and speed. If you have a speed trend vector, a fast/slow indicator or a three digit mach indicator (I guess not on a 737 NG!), then use them for speed trend. The mach indicators third digit serves as a trend because it is so sensitive, not reading the actual mach but checking whether the speed is going up or down - though I am sure you don't have this one on the NG.
Knowing the approximate power settings for the approach is a great help especially if you are inexperienced - at least it saves you from the large thrust changes.
Have fun at the check ride - smile at the instruments and they will smile back to you
