Blue smoke on start and weak acceleration on 01 Honda Civic?

I have a 2001 Honda Civic that has had oil leaks since I bought it about 3 years ago. Because I knew about the slow leak I've been good with keeping an eye on that and adding more when I need to as well as getting a complete change every few months. Recently I noticed blue smoke coming from the tail pipe when I start it in the morning and when I leave work. I've been to a garage that suggested a new cam seal plug and head gasket. Because the gasket was the cheapest I obviously went with that first but now, I'm losing twice as much oil and the car is VERY weak to accelerate to the point where I have to turn my AC off just to keep moving and will blink check engine light if I try to accelerate too fast (Even with the AC off). I've asked around to other garages who have said it could be the valve seals OR the piston seals. I'm lost and don't have much money to spend!

Blue smoke indicates burning oil. Black smoke indicates too much fuel. If this happens after sitting overnight then it is most likely a case of oil slowly seeping into the cylinders. If the smoke declines after a few moments it probably is not the picton rings. That leaves the seepage source being the valve guides, OR, a head gasket leak.

Try this. One morning start your engine and turn it off shortly after the blue smoke subsides. Next remove and examine the spark plugs. The tip of the spark plug from the cylinder with the leak will most likely appear to glissen or be wet, OR be black with burnt carbon deposits.

If this does identity the cause then know this. There's no real fix for it that comes in a can.

Oil smoke on startup is usually bad valve seals. Once the catalytic converter lights off the oil smoke usually becomes too thin to see.

The immediate concern is for the engine performance. The blinking check engine light means serious misfire and doing it under hard acceleration usually means weakness in the ignition system. It appears your 2001 Civic uses Coil On Plug, narrowing the problem down to the plugs (or possibly a bad COP, but that is rarer.) Replacing the plugs would be a good move at this point, especially in light of the oil burning.

Get a new PCV valve and change it. Should help reduce internal pressure. Valve seals would help with the morning blue. Pull the codes to see which cylinder is missing. P030x. An 01 prob has COP so diag it with swapping them.

Sorry for the bad news but you do not have enough money to repair this because your valve guides and cylinders and piston rings are probably worn to the point that the engine needs to be rebuilt

Blue smoke is an indication that oil is getting into the combustion chambers. Lots of reasons such as the engine is dying of old age.

Blue/Gray Smoke: Blue/gray exhaust smoke is an indication of oil burning in the combustion chamber. These are possible symptoms and causes:

Valve Seals: Leaking valve seals will cause blue/gray exhaust smoke.

Valve Guides: Excessive clearance between the valve stem and the valve guide allows oil to leak past the gap into the cylinder.

Piston Rings: Worn or damaged piston rings will cause blow-by, resulting in blue/gray smoke.

Worn Cylinder Walls: Worn cylinder walls cause blow-by, resulting in blue/gray smoke.

PCV System: A stuck closed PCV valve causes excessive crankcase pressure, resulting in blue/gray smoke.

How
many
million
miles?

Have the valves adjusted. Have the valve stem seals replaced.

Blue smoke is good

The mechanic that said it was the cam seal and head gasket wasn't helping.
if the smoke goes away after 5 minutes of driving it's the valve guide seals.
If it smokes under acceleration after driving for 15 minutes the piston rings are bad.
Have you been getting the valves adjusted every 60,000 miles? All Honda motors have a common issue with tight valves and if they are out of adjustment and tight you get a loss of power and a rough idle.
Te valve adjustment has nothing to do with burning oil though.