2002 cadillac wont accelerate to 70 mph - 2000 Cadillac DeVille

It needs connecting to a Scan Tool and checking for Diagnostic Trouble Codes. That will tell the mechanic what the issue is.

check manifold gaskets for leaks, check vacuum hoses for leaks. check the transmission oil condition ( auto tranny)
if it is an auto then it may require the pan to come off and check for mechanical fault.

Have your Mechanic check the steering tie rod bushings, and linkage. Sometimes it's just an adjustment to the steering box play(located at the end of the steering wheel shaft), - easy to fix.

Next time the engine light comes on, get the codes pulled and get back to me

Its possible you have a damaged wheel. A wheel could have been damaged/warped when the first set of tires were replaced. Spin balancing machines "balance" the tires at about 30 mph. Generally that covers things up to 60 or 70 mph on the road.
You may need to have the wheels checked with a gauge to see if any are warped or bent.
Make sure the vibration is not affected by engine torque or other factors. Does it matter if you are coasting at 70 mph or pulling, etc.
A miss-fire in the engine could be transported thru the passenger compartment and cause the same effect, but the check engine light should come on.
If the problem started right after the new tires, I would look at the wheels first.

Since the transmission in this car is electronically controlled the first thing that must be done is to have the transmission control module scanned for fault codes, that is numero uno.

Sticking throttle plate maybe , clean with carb. cleaner .
2002 Saturn high idle at red light
007 Idle Speed Problems Scan Data for Idle

Start by having the code pulled for the check engine light and go from there.

I'd guess you have a tire out of balance, it sets up a harmonic around 30 to 35 mph, and at twice that speed, or 70 mph. I suspect if you drove it at 120 to 140 mph, you'd find another one. Could also be a tire that has a belt that has failed.

How many miles on this set of tires, did you get a lifetime road balance on them? Hazard warranty? How soon before you need new tires. Need to take all of these into account, but need to correct the problem before you have a tire failure and a wreck.

have you tried having your fuel injectors cleaned on it. the engine might not be getting enough gas at take off. it might also be that your gas tank fuel pump or the other fuel pump going bad