How should our glorious civilization incorporate two neighboring fallen empires?

This will be fairly hard to do. First of all, you trigger the awakening process by getting over 50k fleet power, but this then has a MTTH of 50 years (i.e. the monthly chance of triggering an awakening is 1/600). This is halved if your fleet power exceeds 75k. So you can fairly safely build up to 74k fleet power.

If you were to conquer a fallen empire world, the MTTH gets cut by 90%, so it will happen within a few years of game time.

Also for a fallen empire to awaken, their total fleet power must be above 40k.

As such, your best bet would be to declare war against the stronger empire, annihilate their fleet, but do not conquer the worlds. Keep them in check, then declare war against the other empire, destroy their fleet and then conquer both their worlds. This will be very hard even with 74k fleet power and it is pretty much guaranteed that another fallen empire in the galaxy (if there is another) will awaken very quickly.

More information: Stellaris Wiki on upstart awakening

On top of Dulkan's fine answer, I would like to add that until war breaks out with those Fallen Empires, there is one thing your should work on with high priority:

Improve your energy balance

At some point, you will need to quickly increase the size of your Navy, to maximum size. Given your current 344/561, that would be a near doubling.

Given that fielding your Navy already causes a negative balance right now, that will be near impossible to pull off. Even a reserve of several thousand energy credits will quickly disappear facing a monthly deficit of 100 energy or more!

You wrote you are already working on it; Good!

There is no such thing as too much energy.

If you ever find your energy maxed out with a healthy monthly surplus, just terraform some planet.

Claim aggressively

I also noticed you have several unclaimed areas next to your borders, and you have +5 influence every month. Therefore I would suggest to build a couple of Frontier Outposts to claim those areas.

This will increase your research and minerals income without increasing research costs. A quick and easy gain!

Start short wars

Also, there are several smaller empires next to you. Why not attack one of them with a very limited war goal of one planet and the destruction of nearby Frontier Outposts?

The main goal of those short wars is to open up new areas for your Frontier Outposts and Colonies, which results in an increased research income.

The truce after winning the war can be used to scout their planets, and temporarily access the unclaimed areas beyond their area.

In practice, your neighbors likely have a defense pact. Still, their fleets usually can be defeated one by one if they are on opposite sides of you.

Swiftness of execution is key here; You don't want the war to last long. Even White Peace is acceptable once you destroyed those Frontier Outposts with your fleet.

Those skirmishes usually provide a nice boost to the weapon systems you haven't specialized into yet - lasers and cannons.

By the way, why do you tolerate those pirates / drone miners / other aliens in your area? I see are two red [!]s in your area - Get rid of those filthy xenos! [EDIT: Apparently - see comments - those two are too strong. The weak ones have been eliminated :) ]