Plan B: If you have 4", you can slide in 3" and still be able to snake. Even though the cutter will be smaller than the pipe, centrifugal force will make it scrape the walls of the pipe.
Do you think this will still work given the amount of offset between the pipes? (approx 4" - see photo.)
Probably it would not work. I'd not try to snake it with the offset because the cutter may hang on the gap & get stuck or leave the pipe. A four inch offset is big.
---------------------------------------------------------------------------------------------------------------------------------
Money is hard to earn, so perhaps we can help you save some to spend with your family instead.
Since your house is still draining, don't treat this as a panic situation. Don't rush into a repair. Check into options and costs at your pace. Treat is like an experience of learning & exploration, not of stress & anxiety. (Life has enough of the latter already.)
The sewer can still be snaked from indoors. Or-
Perhaps investigate installing a clean-out somewhere else outdoors. I installed one on the driveway wall of my house for outdoor snaking. It ties into a soil stack inside. It is an easy shot to the city main. No digging required.
If you do this, you might be able to get away with filling the existing sleeve like that of an abandoned well. A plug goes near the bottom and the pipe is filled with grout.
A Possible Repair To Try-
As the mathematician Archimedes said, "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.".
If the ground is at all cooperative, you could try to use a piece steel post, tube or angle to lever the bottom of the top pipe in line with the lower pipe. A 2 x 4 on edge may work, too. Once I borrowed a stop sign post for a lever.
The lever should be long & you may have to stand on something to reach the top end. (Pickup truck bed?) Have a helper watch to make sure the lever is not below the separation. Pull to align, jerking, but not slamming, if necessary. Sometimes you will be surprised at how easily the pipes align. Been There. Done That. (More Than Once.)
Then you can install a 3" sleeve.
Use a rigid sleeve such as pvc pipe since you're not back-filling around the outside of the pipe. (Some concrete levelers can do the back-filling if you wish.)
Cause:
Perhaps the pavement sinking has nothing to do with the offset. It may be a symptom, not the cause. It may be unrelated.
Perhaps a tree root pushed the pipes apart, or ground frost did it if your frost line is low enough (more than 1/2 the way from top of pipe to shift: 30" to the shift.) A washout from something else? Neighbor draining a swimming pool to the earth? A clay slide? An animal digging? Simply a bad pipe connector that rusted away.
I'm not ruling it out, but there are a zillion possible causes other than the bottom pipe sinking. (You mention no vertical separation, so the bottom pipe does not seem to have sunk.)
I like concrete jacking & leveling. It is strong.
We've had it done in parking lots, loading docks, dock plate cut-outs, factory floors, machine-slab leveling; even a pair of 12 meter tall stamping presses.
It was also used to re-level the slab under the playing field of an NFL stadium where I worked. (Built on a swamp) I have seen 'construction cheats' where piers and columns were "mud jacked" to level.
A few friends & family used it at their homes. No complaints.
I suggest taking your time and exploring options. But most of all, Enjoy This Day!
Paul