Do It Yourself

The advantage of LVLs are that

1. you can put them up 1 ply at a time
2. they hold nails/cut easily
3. the lumber yard will likely size them for you if you ask

The disadvantage is that the depth will be about 1/16th of the span when using 2-3 plies.

The advantage of steel is that an I beam (W shape is what you want, for "Wide Flange Beam") will be about 2/3 the depth of an LVL. The disadvantages are

1. An engineer will likely charge you about $600-800 to size this beam, but will also tell you how to top connect it
2. It will be one piece (fwiw it will weigh about the same as the LVL)
3. You will have to buy a 40' piece, or pay a premium to have it cut down from a 40' piece. (stock lengths of steel are 20'/40')

Note that nobody can properly answer your question from the data given (edit - just notice you mentioned 16' rafters below). You would need to include the span of the rafters and (at least) your location to determine the snow loads and wind loads (edit: and seismic, though it's unlikely to control for this design) for sizing the connections.

Disclaimer: I'm a structural engineer, but I'm not your structural engineer. For a long span like this I recommend contacting someone licensed in your jurisdiction to help you out.

22 ft unsupported seems like a very long span to me, what's that nearly 7 metres?
Sounds like it's getting into the realm of structural enginneering not diy for me.

If you want to save costs you might think aout a "flitch beam", that's 2 wood beams with a steel plate sandwiched in between - the three components are bolted together every few feet. Easier to join to the timbers then.
https://www.youtube.com/watch?v=QWUNd559UQY

I still think you might be more like 10"x2 or even 12"x2 timbers to cover that span if totally unsupported. But might still come in a little cheaper than the i-beam.
Maybe the roof will be very lightweight and no snow weight is expected - but I'm no structural engineer so don't take my word for it.

Other features like corner bracing or canti-leverage, or some other support structure or other feature (like is it the bottom side of a framed gable triangle) might also help.

LVLmight not be suitable, but i think you can get treated "glulam" beams suitable for exterior (covered) use.
https://en.k2-builders.com/what-type-of-glulam-can-be-used-for-exterior/

To attach wood to an I-beam, you could drill holes in the flanges of the beam, then use bolts. There are also fittings that you can slide posts & beams into, with various ways of securing them to the material they hold. If you're not using T-shaped fittings, you could bolt fittings together.

Easiest thing to do is talk to a metal building company. They do carports all the time. Even if you don't have them do it, you could get detailed info on materials.

Any reason the high side of the shed roof can’t intercept the existing garage roof further up the pitch? Then you could use open web trusses which although deeper are lighter and potentially cheaper than a wide flange. Attaching them higher also gives you more clear height with a longer span.