An invisible creature's light source still emits light
Most arguments against the light source still emitting light seem to argue based on the real world's understanding of electromagnetism.
D&D is neither the real world, nor a physics simulator. The rules only do what they say they do.
For all we know, the worlds of D&D follow the emission theory of vision:
[Empedocles] believed that Aphrodite made the human eye out of the four elements
and that she lit the fire in the eye which shone out from the eye,
making sight possible. If this were true, then one could see
during the night just as well as during the day, so Empedocles
postulated an interaction between rays from the eyes and rays from a
source such as the sun.
Using this theory, it could be argued that it is perfectly reasonable for the creature (and light source) to be unseeable.
Example argument: The invisibility effect prevents "eye" rays from hitting the invisible target (and instead passes through them), but the "source" rays from the light source would not be affected, and so would still shine normally on the surroundings.
However, D&D is not a physics engine for discarded theories either.
As such, we only have the "rules only do what they say they do" to determine what happens.
What the rules say
A bullseye lantern description states:
A bullseye lantern casts bright light in a 60-foot cone and dim light
for an additional 60 feet. Once lit, it burns for 6 hours on a flask
(1 pint) of oil.
If a creature is invisible, there is a description in the conditions for what that means.
However, we are looking at an object being carried being invisible (as the rules for a creature were unenlightening -- no pun intended). I couldn't find any specific rules on what an object being invisible meant so we'll fallback to the English definition of invisible:
incapable by nature of being seen : not perceptible by vision
The lantern is incapable of being seen. Baring any other rules that I've overlooked being relevant, that's all being invisible does for an object. It does NOT prevent light being cast around the object.
Jeremy Crawford's tweet
From this, we reach the same conclusion that Jeremy Crawford provided.
The invisibility spell doesn't prevent you or your gear from emitting
light, yet that light makes you no less invisible. The light appears
to be coming from the air. Spooky! #DnD
The invisible creature can see with its own light source
You also asked:
More importantly, can I still see my surroundings based on this light,
or do I become effectively blinded?
So, we know that light is cast, can you as an invisible creature see it?
According to the rules for Vision and Light:
The presence or absence of light in an environment creates three
categories of illumination: bright light, dim light, and darkness.
Bright light lets most creatures see normally.
Again, given a lack of any other rules to contradict this, yes, the invisible creature can see the light cast by a light source they are carrying. (Similar rules for the dim light cast, but this is far enough down the rabbit hole of rules.)
Additional effects are defined by the DM
Rules only do what they say they do, but the DM is free to add any additional effects they desire.
Personally, I find the description that Jeremy Crawford used to be quite amusing, and
believe my players would as well my players did as well.
(A year an a half after writing this answer, this actually came up in one of my games, and my players used minor illusion to make "ghost sounds" and enhance the "spooky" nature of the light being held by an invisible character.)
However, each DM can rule as they see fit, but they should be consistent from one instance to the next, so the players can plan and play their characters in a consistent world.