Find someone that knows what they are doing and do a 4 wheel alignment on it. Go on the high side with the caster. Check both front and rear tire roll-out.

My "first" dedicated drag race car was a 1960 Dodge Phoenix that weighed over 4100 lbs. without me in it, I put a cheater 1959 SS 383 stroker motor (unknown C.I., 4.375 stroke with a 4.410 bore, I think I didn't have any micrometers back then ) in it. It ran high 12.s at 105 MPH at the old Thermal air port drags in Coachella Valley, CA a long time ago It had a set of 5.0x14 inch stock steel rims on it with a set of Penny 8.00x14 7 inch wide drag slicks with probably way to little air pressure in them It would wonder around every time I let off at any speed above 30 MPH, I soon learned that the back had to follow the front as long as I didn't try to correct with the steering wheel
I would put a tiny bit of left turn into the wheel after I let off in the 1/4 mile and let it slow down while the back end sashay around all it wanted to
I've never crashed a car at the track, YET
Not so on street cars
Be safe, don't crash

Don’t stab the brakes at the end of the track. A lot of problems can come from this.

That's all good info, but that car should not be doing that, especially at that power level. Something is physically wrong with it! I doubt if it is driver error.

I haven't been to the track, Ill be going soon, just gotta drop the broke in 446 into the engine bay and wrap up some interior issues..

I posted this question to get some edumacation bout the shimmys and fishtailing ive seen watching other racers, many which went out of control and smashed up some really nice cars. I just want to be prepared. Im going to do what was suggested, make a few light runs and see if I encounter any steering/suspension issues, and get them worked out before I try again.

I replaced my entire font suspension with new pitman/idler, all tie rods ball joints, control arm bushings, strut rod bushings, center link. had my power steerin box rebuilt, welded steering box supports, reinforced the mounts for the strut rods, skid plate and boxed my lca's, 1.06 firm feel torsion bars, steering shaft connection (forget what that was called) and also frame connectors. Ive done an alignment but will redo it once the engine is in.

What I learned from Frank Hawley is that sometimes if the car isn't going straight, it's because the driver is inadvertently moving the steering wheel from the shock of the launch or when they shift. Without even realizing it. An in-car video camera identifies any driver related problems.
Frank also said that if the car gets out of the groove always abort the pass.

What do you expect to run, ET and speed? Sounds like you have done a lot of suspension work on it already. Fast cars can be no fun if they are ill handling. Did you check setback, tire roll-out? My Land yacht had set back issues ( factory quality control ). Once that was addressed, it was a different car at the track.

The first time down the track I set the throttle cable-rod about half pull. Opening up a little more each pass, you learn more about you and the car till you feel easy makeing an all out pass. Very fail safe. I've seen one guy go full out on the first time out, did a giant wheel stand and come down hard, flip and totaled his car, ended up on the roof. The more HP the more unpredictable it is.
Might add, make your first settings soft, engine tune, suspension, then work up to the best, not on kill right off the bat. Learn your combo in steps.

Toe in is your friend

The only car I had an issue w/the shimmys was my wagon. Issue for me was at 4000 lbs it did not like the old centerline 3.5" wide rims. Add the Flaming River quick ratio gear box and it was a but pucker ride.
Added 5" rims and put in the factory gear box from my cuda and the car goes down the track like I'm cruising down the highway. Night and day. Some like the quick ratio box but for me it just didnt wk.

It is now November, what track are you going to go to soon?

On your deal I think your expecting the worst case scenario, which is good but don't be surprised if it goes straight arrow from A to B with no baloney sauce or problems
You could do a hard launch only or a soft launch and then hit it with WOT until you shift it into high gear and then let off or lift some and drive it to the lights Your choice

Your combination sounds to me like it should be in the mid to high 11s, once it's sorted out. That'a lot of weight, and it doesn't sound like the suspension setup is especially drag-focused, specifically those big torsion bars won't help with getting the front end up off the line to help plant the tires.

I don't think a 275/60 is tall enough. The ST-21 is not a high RPM cam, which is why I think it needs something in the 29"+ diameter to keep the revs down with the 4.10s on the top end (assuming 1/4 mile, not 1/8).

If the suspension & steering aren't sloppy, it should drive fine on radials.