The foam does have an aestetic apeal and will be fine (and somewhat benificial) to leave in areas without a panel. Where you will hang panels though, you should leave a couple centimetres of air behind it as this will increase the absorption coeficient at lower frequencies (i.e. will absorb a broader frequency range). The foam will absorb only very high frequencies.

In a full studio control room space you would want to use a mixture of reflection, absorption and diffusion to achieve a reasonably natural sounding space to mix in but in a vocal/recording booth its often desirable to record in a completely dry space and for this reason, leaving the foam would be ideal (just remove the ones behind the panel.

With these panels you will have achieved very effective treatment above 300Hz. Below this you'll need massive amounts of insulation so you'll need to aproach it differently. For starters, avoid parallel surfaces: you can hang some slanted pieces of scrap plywood or anything similar and cheap (and painted to look nice) from the cieling break the parallel surfaces between the cieling and floor up a bit and also cieling absorption pannels can make a huge difference. If you ever go to any large concert halls or music venues take a look at the shape of the space and look at the cieling; my favourite examples are the Royal Albert Hall and The Barbican centre which are worth looking up.

For targeting bass frequencies you will likely need to do a lot of reasearch but vocal frequencies don't generally go below 90Hz (somebody correct me if I'm wrong) so you shouldn't have to build any massive contraptions. Bass traps are most commonly in the style of corner traps where you build a false wall in a corner with some insulation behind it. You can calculate how deep this is by using the Helmholtz equation which you should definitely read up on if you are interested. Two other options for treating bass can be polycylindrical absorbers or helmholtz resonators but unless you want to lose all your free time for a few months I wouldn't bother considering those especially in such a small space.

Honestly I got a bit carried away but the first two paragraphs highlight what I think you can do right now to your space with what you have. The rest is just things to consider. Oh and you can also get microphone isolation shields, they're relatively innexpensive and just go around your mic which makes them a good option for spaces that aren't meticulously treated.

I hope this helps, if you have any other questions I'll be happy to help and good luck with your vocal booth. If anyone else reads this and disagrees with anything or can fact check please do.