How to counter Protoss's Forge Fast Expand as Zerg?
Solution 1:
You essentially have a few options if you scout a forge fast expand.
Try to out macro them (Defensive) You can play the long term macro game, take a 3rd and out produce him. Because he is going for an economic opening it means that he won't have a strong army so you will be free to drone hard like you say you are. However it sounds like you might be missing injects and not having enough larva to fight the protoss army. Either work on this mechanic or drop a few extra macro hatcheries so that you have some extra larva that way.
Constantly be scouting them so you aren't caught unaware of what they are doing. If you see a stargate you know they are going some type of air play and should be getting appropriate units to match once you know what they are getting (ie. know the difference between getting mutalisks or corruptors), or maybe they're just rushing carriers/mothership. Make sure you scout often because knowledge in this game is vital.
Pressure/All-In(Offensive) There are several kinds of pressure/all-in strategies that you can do if you scout a force fast expand.
You have your traditional baneling bust which will work if they don't have enough defense. You follow this up with lots of zerglings and destroy their workers.
There are variations of a Roach Rush designed to kill/pressure the protoss player if they get too greedy.
You can also use mutalisks to pressure their mineral lines because they will have a large amount of their defenses located at their choke point. This will leave you free to kill probes and get you another economic advantage. After they sufficiently defend this, either with cannons or blink stalkers it depends on how many mutalisks you have. There is a critical mass where you can have enough mutalisks that they will one shot cannons and they won't even matter. You should have a transition out of this.
Your late army composition should consist of infestors, zerglings/roaches, corruptors and broodlords if you play a longer macro game where you can get to these tiers of units. Of course your army composition should be reflective of what you see your opponent getting. For example, if they don't have colossi, don't have corruptors.
Targeting sentries will make your army stronger as it is a very gas heavy unit for them and will cut down on their higher tier units if they choose to remake the lost sentries. Lost sentries also mean that they cannot control most of the fight. What I mean by that is that they cannot force field off parts of your army during an engagement thus making your army weaker.
If you let a protoss get up to an optimal army composition, they will crush your army. A maxed tech protoss army will generally beat a maxed tech zerg army. You may be able to remax quicker and overwhelm them, but proper control from the protoss player will cause you to lose. A combination of storms, colossus, force fields, and archons with some zealots and stalkers plus a mothership will decimate any zerg army.
Solution 2:
Zerg is a very "reactive" race. If you scout that he is going for a FFE then typically there are 2 ways you can combat this.
Roach Push
While not very common anymore: After your spawning pool get a roach warren and go for a really early roach push with around 7 or 8 roaches. You could also try Stephano's build which involves getting 3 bases and getting maxed out 200/200 supply made of roaches.
Take 3 bases early
Take 3 bases super early. After starting your second go for a 3rd and focus for the first part on saturating those bases.
Never 1 solution!
As for your concerns about the Protoss taking out your third: There is no ONE solution to the problem. It always depends on the situation at hand.
Continuous Scouting
The key is to continuously scout. Use overlord scouting as well as sending a ling to the front of his base and see what kills it.React according to what you see!
You can then start planning the production buildings accordingly. If you see him having a massive army starting to emerge STOP droning and go for the composition you were trying to get. You should have a good amount of saturation at all of your bases by this time assuming you are on point with your queen injections.Don't be afraid to swap armies
If he does move out you have some options. If you have a perfect unit composition for what he has direct confront and surround with lings! Swap armies with the Protoss if you can. His troops will most likely be more costly and he is off of less bases then you. As Zerg you can almost instantly recreate the army you lost.
Counter Attack
The other option is that you counter attack. Take your force and wait for the Protoss to get out of his base then run in with your force and take everything you can out. The Protoss is then forced to either retreat and attempt to deal with your force or continue with his attack. At this point it is a base race and you should have more to kill then he does.