First off, I suggest you review the
AC sticky I wrote.
The first thing you have to consider right now is what kind of sealer you put in the system. If you used a hardening sealer (aka "professional" aka "seals leaks in metal", then your system is probably done. Those sealers can plug the orifice and destroy the compressor. The residue is essentially impossible to remove. They are like an epoxy in that they set hard. The catalyst is moisture so if there is any moisture in the system at all, you will have tiny particles of hardened sealer floating around in there, grinding away at your compressor and plugging things up. You may as well have thrown in a handful of sand. The ONLY excuse for using a hardening sealer is when there is a known small leak in the evapourator, the labour for replacing the evapourator exceeds the value of the car (or at least, the value of having AC) and you are willing to write off the AC system if it doesn't work. Hardening sealer should ONLY be used AFTER the accumulator is replaced and the system thoroughly evacuated with a deep vacuum for several hours on a warm day. Not with one of those cheap air-compressor powered vacuum pumps either -- you need, at a minimum, a single stage electric vacuum pump. (Harbour Fright actually sells an acceptable pump for a reasonable price.)
The other kind of sealer is supposed to swell seals and stop "soft" leaks like o-rings and seals. First off, nothing can stop a compressor seal leak except a new compressor seal. Second, it's unlikely they will even stop an o-ring leak. OTOH, seal swell additives probably won't destroy your system either.
You mention a black goo coming from a line. In AC circles, this is commonly called "black death". Usually the result of moisture contamination in the system, but sealers can certainly contribute. Many AC oils are hygroscopic, that is, they have a great affinity for water. Couple that with metallic residue from compressor wear and the resultant sludge gums up the system, plugging the orifice and/or screen.
Best case scenario, you are looking at a major overhaul here. Forget those "flush in a can" kits -- they do not have the volume and pressure to clean out your system. A high-volume flushing machine might work for some things like lines and difficult to access parts like the evaporator. There is only one DIY flushing system that I would even consider,
the Hecat pulsator system. It's not cheap ($400) though so you might be better just to replace the parts, especially since it sounds like you have to replace at least one of the lines anyway.
Some parts cannot be flushed, period. The compressor is one, the parallel-flow condenser is another. All lines must be separated and all components flushed individually. The accumulator/dryer must be replaced anytime the system is opened for more than a couple of hours on a dry day or minutes on a humid day -- and "open" includes "leaked down to 0". Your orifice is undoubtedly plugged by now.
The compressor is a bit of a crap shoot at this point. You know the seal is leaking. Since there is evidence of "black death", the compressor must be removed, the sludge (I won't call it oil anymore) drained, case split, innards cleaned out and inspected, new seals installed. Do you know if it was working properly? IE: did you take readings with a manifold gauge set that showed a normal pressure differential last time you recharged it? If so, there's a chance it is salvageable and might be worth saving.
At this point, assuming you did not use a "professional" sealer, and assuming there is no evidence of dye at the condensate drain (use a compact fluorescent bulb in a drop-light. The LED flashlights and incandescent bulbs are virtually useless) I would replace at least the leaking line, and preferably the others too. If you don't replace the other lines, at least replace the o-rings on them. I would just bite the bullet and replace the compressor. Replace the orifice, accumulator and condenser. Install an in-line filter between the accumulator and the compressor. When installing the new compressor, drain out the shipping oil and install fresh oil. Put a couple of ounces of oil in the accumulator and a half ounce or so in the condenser. Put the balance of a full charge in the compressor. GM specifies PAG oil, but I personally prefer ester. Add an ounce of dye (or buy oil with the dye already in it, which is what I do).
If you used a "professional" sealer, then there is no debate: just replace everything or do without AC. You are kidding yourself and wasting your money if you think you can get it all out again.
Once everything is all buttoned up, pull a hard vacuum overnight in a warm location. Ideally, over 70°F. The vacuum may tell you if you still have a leak (not always -- some things will leak under pressure but not under vacuum and vice-versa) but more importantly, you are boiling off any residual water vapour hiding in the system. Literally -- a hard vacuum will lower the boiling point of water to less than room temperature! Run the pump for about 1/2 an hour the first time. If you still see vapour coming from the pump discharge, leave it run longer. After leaving the system to boil for several hours, run the pump again for another 1/2 hour. If you still have visible vapour, leave it for another few hours. The pressure may climb slightly the first couple of times, as the "steam" fills the system, but eventually you should see no pressure rise after sitting an hour.
Purge your fill hose! Easy if you are using a manifold gauge set, but may not be possible with a cheap fill kit from Wally World. Fill with liquid (ie: invert the can before opening) to get the most out of it. Do not overfill! The weight of the contents are on the side of the can. Add 'em up and compare to the sticker on the AC system. For the first can, the vacuum will suck most of it in before you start the engine. You can warm the can (pot of hot water, lay it on the warm engine) to get the most out of it.
Sorry, I'm sure you really didn't want to hear any of the above, but them's the harsh facts.