From "Range"
Once a spell is cast it's effects aren't limited by its range, unless the spell description says otherwise (PHB p203).
Thus no, you don't have to maintain contact (unless the spell says so) when casting a touch concentration spell, just like you don't have to remain in range to concentrate on other spells, as the range is only a limitation when the spell is cast, not when sustaining it.
Can the GM ban references?
Yes. Specifically, "the D&D rules help you and the other players have a good time, but the rules aren't in charge. You're the DM, and you are in charge of the game." (DMG p.4, "The Dungeon Master," emphasis in original.)
That being said...
This is strange, as presented. In nigh-thirty years playing I've not run across a GM who disallowed looking up references. Not during your turn? Sure. Not at all? Weird.
(When running a table that operates under time constraints I will often skip a player who's not "ready" for their turn in combat. This tends to fall harder on spellcasters than on martial types, and harder on new players than on experienced hands. Of course, this is articulated at the beginning as a standard of play.)
But according to your edit, that's not what's going on here.
Your GM seems to be basing this restriction on two ideas: the character needs some time to be able to "fully" use their class features, and the player's skill should impact the character's abilities.
The character can't use their class features--new spells, in this case--until they're "broken in."
This has no basis in the rules. In short, "Beyond 1st level" (PHB p.15) tells us that when your character earns certain XP they gain a level. The class descriptions tell us what new things the character can do when they gain a level.
@Kryan's answer has the right of this: the two new spells your wizard now knows by dint of increasing a level represent the work that character's already done to learn new spells, not some new task the character needs to take on.
(As an aside, do fighters only get part of their Ability Score Improvement or feat until they've sufficiently proven themselves at their new level? Do druids pop out of Wild Shape suddenly because they're not yet well-trained in a new form?)
Where there is some support for something like this is in the question of when a character earns XP or can gain a level. Adventurer's League rules, for instance, only allow a character to gain a level when they've completed a long rest or at the end of a module. That is, if killing goblin three of seven in an encounter would put you at a new level, we usually don't stop combat to do it then.
This is also subtly achieved by GMs who award XP at the end of sessions, or by "milestoning."
Again, though, once you've got the level, you get all the class features that come with it, full stop.
Player skill = character ability. Charging your character an action for you to look at your character's spells is... insane. That's a huge hit in the action economy.
This seems to stem from an idea that the player's ability to memorize everything redounds to the character's ability to perform in the fiction.
The idea that player skill should be important is an old one, and has plenty of merit to it, I think.
But this is an incredibly ham-handed way to bring player skill into the game.
(Again, I've got to wonder if your GM tries to stab the fighter-player just to see how well the player reflects DEX 18?)
"Shouldn't I be allowed to reference my spells...?"
This is hard to answer: there may be things going on that we're not aware of. What you should be able to do, without question, is talk to your GM about what's going on. Ask them what purpose they see their rule serving. Ask what they're hoping to achieve. Describe the difficulty it's causing you. Ask them for help playing your character well at their table.
(This all assumes the very best of your GM: that there's a good reason, poorly articulated, for this and that they're interested in helping you play your character. I hope that's your situation, rather than the other one: you're sitting at a table with a petty tyrant who hates spellcasters and is actively trying to make play difficult for their players. In that case I suggest you find a different table.)
Best Answer
Many of the concentration spells in 5e revolve around some kind of continuous lockdown, buff, or damaging effect. For example, Bane, Bless, Spirit Guardians, Barkskin, Globe of Invulnerability, etc.
The goals are to prevent spellcasters from being able to apply multiple effects at once and to force players to think carefully about their spell choices, and combat strategy.
For example, in a world without concentration, a cleric could just bless his whole party, bane a bunch of enemies, and run straight into the fray with spirit guardians so that every creature now makes the save with a 1d4 penalty. Heck, he might as well throw a shield of faith on himself too. Wizards could cast hold person, witch bolt the poor paralyzed thing with advantage, and continually zap it over and over again with no fear of it running out of range. Yes, doing these huge combos of spells can cost a lot of spell slots and actions, but it makes smaller, more intimate combat less tactical. Plus, it gives the affected creatures no way to escape their predicament without liberal use of dispel magic and counterspell.
And these problems apply to PCs too. Save or suck spells are only fun when you aren't on the receiving end. Imagine an evil cleric doing what I described above against your party, and it feels really cheap to fight against.
With concentration, you force players to make choices about which continuous effect suits the situation best. It introduces excitement in that these spells can be ended early if the caster fails a concentration save, and it doesn't impede your ability to cast classic one-and-done spells like Magic Missiles, Fire Ball, or Cure Wounds. It also has the benefit of working reasonably well both for and against the PCs.
With that being said, the limited number of higher level spell slots means that concentration isn't as necessary for high level spells. Since a full caster will have six 6th-9th level slots compared to sixteen 1st-5th level slots at 20th level (even fewer high-level spells at lower levels) you can consider dropping the concentration requirement even for continuous effects if you increase the spell level to compensate.
When you introduce custom spells into your campaign setting, I would follow the following guidelines:
When it comes to homebrew material, a dose of playtesting will always be required for optimal balance. You should keep an open dialogue with your players about potentially changing the effects of the spells that you add as you see the players using them in game and discover what their impact on the game really is.