Well you can go to auto zone and rent a compression gauge for free, all you have to do is pull the spark plugs (disconnect the coil harness clip so the coil isn't trying to fire while you're doing it) and screw the gauge in place of the plugs. I guess that is probably a lot further down the troubleshooting list.
If this were my car, I would do the following:
1) replace oil sending unit. By the way, I just did another sending unit over the week and NAPA sells one that is a factory match, use them if you can, it was less than $10. Coat the threads with Hondabond/Permatex Ultra Grey silicone.If the light goes out, good, you're done and that's all it was. If the light continues to stay on while you know there is a full 4.5-5 qts in the engine then you know that the pump is truly not generating enough pressure. Don't freak out just yet, you MAY not need to replace the oil pump-
2) Pull the valve cover off the top of the engine. It is held down by 5 10mm bolts- er nuts on your LX- and set the cover aside. Should take just a couple of minutes to pull it off. This is a bird's eye view of how bad the inside of the engine looks. If it is all sludged up from not being maintained like you said, then your pump or pickup may be clogged pretty bad. At this point I would then drain the oul and proceed to...
3) Pull the oil pan. This is time consuming and tedious but not particularly hard. You need a 3/8 ratchet and a 10mm socket for the pan bolts. You also need a 14mm and a 12mm socket, a few extensions, and a 12mm and 14mm wrench. You have to remove 5 bolts from the chassis crossmember, then it will wiggle out. It has an exhaust hanger through it, but it slips out. Then you have to remove the exhaust downpipe. To do this you use the 12/14mm (can't remember which one) socket and extensions and remove the 3 nuts at the exhaust manifold. Then you use the wrenches to remove the two spring bolts from the test pipe (catalytic converter end of the test pipe). Then there are two bolts holding the downpipe to the back of the block, those two have to come out as well. Finally, remove the three 12mm bolts holding the transmission shift selector cover onto the transmission, then take out the three bolts in the flywheel cover. I know that sounds like a lot, but it really os all just about gaining access to the pan, its a total of only 13 bolts/nuts I think to get to the pan.
Dropping the pan is pretty simple- just remove the 12-16 bolts/nuts (can't remember how many) and it will drop right down. It's best to let the car sit over night before pulling the pan so that oil isn't dripping down the engine like crazy when you go to put the gasket on. Anyway, sometimes it's a PITA, sometimes it's simple to get the old gasket off the pan. I use a razor blade and brake cleaner on blue shop towels. Make sure you scrape the hondabond off the block surface too, there's usually a very small amount of residue there.
Now, the real reason I say to pull the pan in this oil pressure troubleshooting is to check out the pan and the oil pickup screen. When the pan is off, you can tell if the pan is full of sludge, the pan will be lines with all kinds of thick slime. You can also look up in the slot of the oil pump pickup and see if it is clogged or full of crap. If the bottom end is really messy, I can tell you there's a good chance the pump is already shot. But if the screen is clogged pretty good, the pressure issue could just be the pump not being able to generate pressure because it was bogged down by the crap on the screen. You can drop the pickup if I remember right, which would let you run cleaner through the other side of the screen to clean it off. You may just be able to clean it without removing it and then put the car back together to see if cleaning the screen and pan out took care of it.
If the screen is fine though, odds are the pump does have too much wear and would need replaced. It's $172 according to autozone, and at that point, you have to decide whether the core engine internals are worth putting that kind of money into. I guess knowing that there is good compression and you aren't leaking past pistons would help in your decision to proceed with replacing the oil pump. To replace the oil pump, you have to keep the valve cover and oil pan off and then remove the alternator, loosen the power steering, remove the crank pulley (requires special tool), remove the engine mount, remove the upper and lover timing covers, remove the timing belt and balance shaft belt, the timing tensioners, water pump, balance shaft sprocket, crank timing sprocket, and then you can take the oil pump off. The oil pump is essentially the entire lower faceplate of the timing end of the engine, it's not a light project. And if you can tell just by pulling the valve cover that the engine is sludged horribly, I would drain the oil, put ATF and seafoam in and run it for awhile, and do a couple of flushes to break the sludge loose before you possibly have to put the new oil pump in- try to get the rest of the engine clean so if it's going to make a mess from breaking it all loose, you won't be doing that after you put the new pump in.
Basically, as you can tell, you almost might as well do an engine swap- and you have to ask yourself if it's worth the grief. I have a 96 Accord that I love to death, I have worked the whole thing over and it's a pretty solid car, but it still just has a lot of little problems- lots and lots of little problems. I just bought a 98 Accord yesterday to replace it, I told myself that the money I was going to have to put into my older car (and do the labor) was better spent just buying a slightly newer ride that I wouldn't have to wrench on.
Basically I say that to suggest to you that you might be better off selling it and picking up something newer on craigslist. Here's a list of how much you could end up spending, parts alone, diving into that project:
Tube of Hondabond: $10
Oil Pan Gasket: $20
Valve Cover Gasket $30
Oil sending unit: $ 6
Oil Pump: $172
Oil Pump Seals
(crank, balance shaft) $15
Water Pump gasket $ 2
Oil, ATF and seafoam $ 60
Then of course "while you're in there" you'd be tempted to drop $150+ on timing components for maintenance. So you're talking $315-500+ dollars you could end up dropping. So, if you could sell your car as is for like, $1500 and put another $500-800 with it and try to land a newer one with less miles, it may save you a huge headache.
Sorry to take a pessimistic turn with this post, it could just be the unit, or it might just need the screen cleaned, but I said all of this only to help educate you so you can make a better informed decision on what could be the problem, what might help, and what you may be getting into. Good luck and keep us posted! Btw, did you ever get to the bottom of that overheating issue you were having?
