Overloaded circuit on ungrounded outlets
Not a lost neutral

A lost neutral in an MWBC is easy to test for.   Just remove all loads and plug in two 240V-tolerant loads, one into each half of the MWBC.  (watch it - the appliances may see voltages as high as 240V!) If one leg has higher voltage than 120, and the other has lower voltage than 120, and they total to 240, then you have a lost neutral.   

However, from the sound of it, that's not what you have. It sounds like you have a lost hot wire.  

Phantom voltage is confusing everything

Also, you are running into a phenomenon called "phantom voltage".   If several wires are run parallel for a distance (like any wires in cables, or cables next to each other), the wires will have capacitive coupling.  If one wire is disconnected/floating, and a voltage will appear on it, induced from the other wires.   This voltage has no strength.  It can move the "needle" of a digital voltmeter; those are so sensitive they will read the phantom voltage.  It cannot move the needle of an old analog meter in most cases.  Apparently, it can dimly light certain cheap LEDs.  

Phantom voltage is never full voltage.  It is somewhat less. For instance if your mains voltage is 121 volts, and you measure 109 volts with your DVM, that's not "meauring 120V".  That's phantom voltage.  

Phantom voltage means the wire is disconnected between here and the source, and it runs alongside or in the cable of live wires. 

It appears your lighting MWBC is inducing phantom voltage onto your outlet MWBC.  That's interesting, because the two hot wires in a MWBC are supposed to be opposed, and you would expect capacitive coupling from them to be equal and opposite.  In other words, to cancel each other out. **I suspect this is because one wire of your lighting MWBC has a wire break.  

Tied neutrals

So far, you may be thinking "well, all the evidence I saw of a tied neutral, may just be phantom voltage.” That may well be.  But a tied neutral is a very serious defect.  It can result in that neutral overloading.  And what just happened here?  You plugged in a heavy load, and the circuit quit. 

So as you search this wiring, you must separate any neutrals between the 15A circuit and 20A circuit.  If you have any smart switches in the 15A circuit stealing neutral from the 20A circuit, get rid of those.  You can't have that; it's illegal and it will painfully complicate troubleshooting.  Normally I would say distinguish them with tape markings, but they are different sizes so that should be obvious.  It is OK to tie safety grounds, but you say you don't have any. 

Follow the dead hot/phantom-voltaged hot back to the panel

Let the phantom voltage be your guide. The wire break is either at the first failed outlet in the chain, or last working outlet.  I'm using the royal meaning of "outlet" to include any point of use, including switches.  

One more thing.  The single likeliest cause of problems like this is backstab connections.  Those are uninspectable.  The only way to inspect them is destructive.  Push the release tab if you can, and then twist and pull the wire out. (Never cut. Wire length is precious, but our entire point here is to inspect the side of the wire for spatter and spallation marks from arcing).

If you found tiny arcing marks, you probably found your problem.  Welcome to backstabs.

Like we say, don't use backstabs at all; but if you're tempted to just jab it back in, don't.  The backstab's tiny spring is now weakened, and if used again, will only cause problems like this.  The backstab has 4 springs and cost 50 cents, what do you expect from it? :) Use the side screws. 

Priorities

I would focus on fixing the lighting circuit, patrolling for tied neutrals, and then, once lighting is back, focus on the outlet circuit.