...(I was told it is the street friendly tunnel ram but can't find any markings on it to verify that) and 2 500 CFM Edelbrock carbs...
Street-friendly tunnel ram?

If you want more information on those, they are listed in the catalog, right next to population-friendly hydrogen bombs, and screen-bottomed boats.
1. The wee Edelbrock SBC tunnel ram has a recommended power band of 3,500 - 7,500 RPM. See that bottom number? 3,500? At what RPM does your engine idle? I'll wager it is well below 3,500 RPM. Any more questions on why it falls flat on its face, when you go WOT from idle?
2.
Go read this post. Let's pretend, (yes, it is a helluva stretch, but let's make the numbers easy) you could actually achieve 100% volumetric efficiency with your 350. It is impossible, but just play along with me, here. An engine of that bore and stroke, at 6,000 RPM (and how much time does your 350 spend up that high), your 350 requires a 'whopping' <cough> 608 cubic feet of air flow, per minute. And you have 1,000 CFM of carburetor? Any more questions on why your engine falls flat on its face, when you go WOT from idle? Here, let me help you with some real world facts. You are going to be mighty good to get as much as 75% volumetric efficiency out of your engine. Suddenly, your engine has an air flow requirement of 456 CFM. Please keep in mind that you have over double that amount.
I just don't like the significant stutter it has if you jump on it.
Ditch the tunnel ram and twin carbs. See how easy that was? Depending on the cam profile you are using (RV tells us nothing), bolt on a 600 CFM carb on a dual plane intake, and the combination will tend more in the direction of ripping your head off, from idle. Nope, you're no longer going to look the part, but in my book, running the part is the important role.
I know back when I was messing with cars a lot more I would think the above problem was too much carburetor.
It's a funny thing about physics, the science never seems to change.
I have done some searches and found some smaller Holley carbs (390 CFM), but they are pretty expensive and I am not sure that will fix my stutter.
No, they won't. They might be a better Band-Aid on your problem, but the end game is that your intake manifold is not designed to work off-idle.
I really like the looks of the tunnel ram and would like to keep dual carbs on top.
In that case, live with the massive hole the motor falls into, when you stab it.
As rcnurd mentioned, you might be able to cover up some of the stumble with a lot more accelerator pump shot, in an effort to help wet the intake, but the bottom line is that the tunnel ram will never work as well as a dual-plane, or extremely short-runner single plane intake, on a street car.
I've explained this umpty-gajillion times before, but the secret to throttle response is tailoring runner lengths, plenum sizes, and RPM ranges to engine size.
When an intake valve opens, that particular piston is making its way to BDC, correct? That action creates a low pressure area, above the piston. If the intake valve opens really early, then that low pressure signal has not had much time to build in energy. Either way, when the valve opens, there is now a low pressure signal moving up through the runner of your intake. That signal is what allows a carburetor to work, because when the carb sees a pressure differential existing between the area above the carb and the area below the carb, that is when it allows air to flow. And that air flowing is what picks up atomized gasoline, and delivers it to that cylinder that is trying to fill.
Go get a balloon from a party store. Blow it up. Not all that hard to do, is it? Now, go get a hot air balloon, the kind people hang gondolas from, and fly about in. Now, using the same method as before, blow up the big balloon. Huh? What do you mean you cannot do it? What are you trying to tell me, your wee lungs cannot make as much difference on that larger volume balloon? Gee. Imagine that.
Put a straw in your mouth, then hold your hand up, 6" in front of the end of the straw. Blow, as hard as you can. Feel that air? Now, stand at the front door of your house, and have someone stand at the back door. Do it again, blow on that straw as hard as you can. Did your helper feel anything?
Now. After you've considered all that, let's go back and look at your intake. Do you reckon the reason your motor stumbles is because of those long manifold runners, and that comparatively huge plenum?
If you want throttle response, use small valves, in heads with small runners, use intakes with short runners and small runner cross-sections. Do everything you can to preserve that vacuum signal's strength, as it travels from the valve to the bottom of the carburetor boosters, because that vacuum signal is what is going to give you that snappy throttle response. Start increasing runner lengths, and the carb is going to lag behind the engine. Start increasing runner cross-sections and that vacuum signal is going to sputter out, and die on you. Smaller is better, every time. If you increase manifold plenum volume, runner volumes, or valve sizes, you need to be sure you are making the equal and opposite changes, to cover up the problems you are creating.
Remember when you were a child, and you were toying around with a garden hose? How did you make the water stream spray farther? How did you make the water stream have more energy? Didn't you cover part of the end of that hose with your thumb?
I just told Zandoz in another carb thread, this stuff is not rocket science. It is actually very simple and very straight forward. So why does everyone insist on making it harder than it really is? Stop trying to over-think things. Pre-school children know how to make a garden hose spray farther, but educated adults cannot understand that stupidly simple principle.
Occam's Razor. Or K.I.S.S., if you prefer.