Squeak, wiggle, squeak, wiggle

My top guess would be the wheels are out of quarter and a rod is binding.

OK.  Is this condition one wrench, two wrenches, or three wrenches to fix?

Three wenches, four hands, and a little luck.

Thanks for the insight.  So I imagine it would be $$$ as well as three wrenches?

Trust you have inspected for bent axle(s)?

I just checked for a bent axle and nothing seems to be out of order there, the wheels all look normal and straight as I hand turn the gears

If this engine has a spur-gear drive motor, did you ensure you oiled both sides of the armature shaft where it goes through both parallel plates that engulf it? The armature spins at high speed and tends to dry out the bronze or oilite bearings it runs in.

Yep oil the armature shaft ends,

The wobble is probably as John stated above, but it could be a bent axle, or dirt on the drive axil between the wheel and frame. You would be surprised at how much carpet fibers, cat hairs and such can get pulled up in there.

As per the squeak, check your roller pick ups. I have a old Williams M1a Mountain squeak like crazy and no matter what I did with cleaning, oil and grease I could not get rid of it. Until one day I pulled it off the shelf to show a friend and the first thing out of his mouth was to ask why the pick up rollers where so flat.

It seems they got jammed up and where just sliding along the rail. I bought some new rollers at a train show that looks like they will fit and I will get around to changing them as soon as I complete the other fifteen projects in front of it.

In any case look for the obvious before you do major surgery on it. 

I put some Labelle's gear lube on the gears a little while ago before playing some golf and the squeak is virtually gone and the wobble is far less too.  I had asked my 93 year old father in law to lube the gears the other day at his house using the Labelle product and I think he may have lubed the opposite side of the six wheels (not where the gears are located).  Hey, he's 93, he gets a pass for not lubing the gears.  That may not solve the problem entirely but it is much reduced in severity.

Regarding D&H 65's post above, I think this locomotive has a "spur gear motor" but I have not had the shell off and have not looked at the motor and as a beginner it's unclear to me exactly what you are talking about but I will do my best to lube the armature shaft and anything else that looks like it needs lubrication.   Last night it was running fairly smoothly for a time but this morning it's still squeaking and lurching.  Certainly a look at the motor and drive is the next step