In a case where you vacuum your items in those airless bags(Space bags), do they weigh more ? Or is it just a feeling because of increase in density due to decrease in volume ?
For any every day life purpose, they weight the same. But there is a difference:
Weight vs mass
Weight and mass are different things. The mass is a measurement of the amount of matter (alternatively, the opposition of a body to change trajectory), whereas the weight is the strength with the Earth pulls objects towards it. The difference is subtle: an astronaut in orbit has mass, but no weight; a scale measures weight, but it transforms it to mass.
Now think of a helium balloon as it goes up. Why does it go up, if it has some mass, and thus the Earth pulls it downwards? Because its weight is less than the air around, so the Earth is pulling from it more, and so it has preference.
Or, in other words, Archimedes principle holds. If you measure it, the balloon has negative weight.
Feathers and lead
If you weight a ton (mass) of feathers and a ton of lead, you will get that the lead is slightly heavier: the air around the feathers is pushing it upwards more than the air around the lead, because feathers take much more space, and thus, more air helps with buoyancy.
A bag full of air
Now we get to your question. We have two rigid identical boxes, one empty, and other full of air. The second one will weight a bit more, and also have more mass. The effect of the air is the same for both, as they have the same volume.
Now, take two soft plastic bag and fill one with air. Again, the mass of the one with air will be greater, but the weight will be the same. Why? Because the one that is empty has also a smaller volume.
Do a thought experiment: take the rigid box full with air and put it on a scale. Carefully open the lid. Nothing has changed, the weight is the same. In the soft bags, the extra mass added by the bag gets exactly cancelled out by the buoyancy (by Archimedes' principle) provided by the extra volume.
All this efects are small. For volumens of a liter, we are talking about differences of 1g. As you righfully note, the density increases, and we tend to feel it as heavier.
The actual weight of the object itself (without air in it) will remain unchanged. Yet before you sucked out the air, the total weight of the thing (object + air inside it) was less.
So it's similar to a balloon case. With air in it, the balloon falls down more slowly than without, because the total weight decreases, and it's only partially due to the increase in size. Helium-filled balloon goes up, because it's total weight is less than the air outside, so despite the gravity still working on the helium-filled balloon, the air around is heavier, so the balloon is pushed up by the air (like oil by water).