Starter problems!!!

Hi. I don't have much automotive experience, but I'm attempting to get my 68 Camaro back on the road after a 15 year storage period. So far, I have removed the heads checked and cleaned everything, reassembled everything (with the help of DennyW regarding my distributor), and I am at the start up stage now.
Here's the problem. The engine cranks over no problem for awhile, but then the starter makes a horrible grinding sound. If I loosen the starter and then retighten it again, it will be okay for a little longer and then the same problem comes back.
I assumed the teeth on the starter gear were worn, so I put a new starter in, but the problem remains. I have inspected the flywheel and it isn't missing teeth and doesn't look worn. A neighbor mentioned that the starter may need to be shimmed, so I have tried to add various sizes of shims with little success.
The engine and tranny were rebuilt, with a new flywheel, before I took it off the road (it has just over 4000 miles on it).
The engine has fired up a couple of times, but dies before I can adjust the timing or carb, when I go to fire it up again the starter eventually screws up.
I feel I'm really close to getting it going, but am really frustrated. If anybody has any ideas, I would love to hear them. Thanks.

Thanks again Denny.
I normally crank the engine until it fires up or the starter craps out. This can be anywhere from 2 seconds to 20 seconds. It starts pretty quickly now, so if it can crank for about 4 secs it will start. You gave me advice about the distributor before and I think the timing is fairly close. I haven't adjusted the idle on the carb yet because I was more concerned about the starter problem.
As for the shims, I've been adding and removing shims and just trying. I haven't measured anything. if you have pictures or detailed instructions on shimming the starter, it would be appreciated greatly.

I agree with Denny that the battery may not be up to par, have you charged it lately. Another thing might be bad connections. Check your connections after an attempt to start, especially at your battery and starter so see if they are getting hot, if so clean up your connections real good and charge that battery if not already done.

Thanks guys.
The battery has been recharged, so I don't think it is the problem, but I will check the cables.
When the engine stops cranking and the starter makes a grinding noise, I notice that sometimes the starter gears are extended and have not meshed properly with the flywheel gears. When I loosen off the starter the gears retract again. This makes me think that the alignment could be an issue
Thanks for that link Denny, it was very helpful. How can I extend the starter gear to measure the space between the two sets of teeth?
Once again, I really appreciate your guys advice and patience.

Is your ignition key switch backing off the cranking position fully,,,,Just a thought,,,,it may be delaying it's non start position.
Other than that,,,do as suggested and pull the solenoid,,and advance the drive gear manualy.
Shim as neccesary.

Also,,,because you said the flywheel had been replaced,,,check the "runout" make sure it's true,,becauise that may be part or all of your problem.

Hi Guys,
Thanks for your advice. I went and got the battery fully charged, and in the process found the wire going from the coil to the solenoid was melted and brittle. I thought this was the likely source of the problem and replaced both wires going to the solenoid. But when I put the battery back in and tried again, I had the same problem.
I removed the flywheel cover and noticed that the teeth on the starter were only just meshing with the flywheel teeth. The teeth had to move into the flywheel by about 1/4", shimming the starter only brings the starter down from the flywhel, which increases the distance.
I'm thinking I may have had the wrong starter on there the whole time. I went to a couple of local parts stores and they think it is the correct starter but weren't fully sure. The starter right now has offset bolts, but there are 3 bolt holes in the block, so I assume I could use a starter with offset bolts or lined up bolts.
If I tried a starter with the different bolt configuration, is there a chance that the gears would move over by1/4" and engage properly? I don't want to buy a starter and have it not work.
The numbers on the block are 3970010 and the transmission is a Turbo 400. Any ideas? It's driving me f***ing crazy. Thanks again

Just a thought--are you using the correct bolts to hold the starter on? Chevys need the proper knurled bolts to align the starter. Another wierd thing I've seen is the ring gear installed backwards! The starter then barely touches the gear, and makes a terrible racket. Aftermarket ring gears (flex plates) often can be installed wrong, the factory ones have a raised lip on the center hole that prevents this.

I've tried all the advice everyone has given, which I really appreciate, but still have no luck. I have tried numerous starter heads, but none fit better than the one I have. I have tried starters with offset bolts and starters with parallel bolts. I have tried shimming the starters.
It seems that when the starter is bolted up, the gears start binding by the second tooth, but when it is loose they mesh well.
I had a neighbor that is a mechanic come and have a look and he thought it needed a different starter, but everyone he tried didn't work.
The flexplate isn't worn and not backwards, but I'm beginning to think that it may be the problem.
Here is what I know; The engine is a 350 LT1 from a '70 or '71 Vette, the tranny is a Turbo 400 with a 2800 hi stall, the flexplate is 14" with 168 teeth, there are three bolt holes for the starter ( I assume for the offset bolts or parallel bolts).
If anybody has any ideas (however stupid or crazy), I would really appreciate it.
My next step is to remove the tranny and try a different flexplate.
Thanks

hey hammer, your saying that when you loosen starter that it seems to be ready to go again for a short time,correct? if this is the case sounds like you have a shimming problem and the starter is moving away from the flywheel and if you continue you will mess up the flywheel.Try this as a rule of thumb/ with the starter mounted take you a 1/8 rod or allen wrench and slide between the starter shaft and flywheel this should be a snug fit or close to it.if the clearance is to much shim just the outside of starter if its not enough shim just the inside of stater and if your working a bb you may have to add a gm brace on the back of starter to help keep it in place 2 styles, also make sure your not trying to start the motor with timing to advanced causing it to kick back makeing starter move. wjr