This is a trickier question than I thought it would be, and there is no definite answer laid out in the rules. To give the conclusion in advance...
You may be able to protect allies from your skin while using Touch-range spells on them.
But you'll need to ask your DM.
Here's the complication
On the one hand, Touch Spells are blocked by the armor of your target. A spell like Inflict Wounds makes a Melee Spell Attack against a target's AC. There's no rule in there that this attack bypasses the AC bonus from armor, so clearly a Touch Attack spell has to get around their armor. However, there is also no rule in place that says you have to contact skin--otherwise something like winter clothing would also provide protection from Touch spells.
Complicating the issue a bit, there are a lot of spellcasters who can wear armor--and to provide the description of Chain Mail:
Made of interlocking metal rings, chain mail includes a layer of quilted fabric worn underneath the mail to prevent chafing and to cushion the impact of blows. The suit includes gauntlets.
In the absence of any rules that say you have to take your gauntlets off to cast a Touch spell, it can be assumed that a Touch spell can work through them. Given that clothing doesn't seem to stop a Touch attack, this makes a sort of sense--the palm of a gauntlet is often made from soft leather...more like clothing than armor.
This leaves us in a rather odd place, from a realism standpoint. Heavy winter clothing may be just as thick (or thicker than) padded armor, but padded armor provides protection from a Touch spell that winter clothing does not. Ultimately, this appears to be a piece of unrealism in the mechanics that is ignored in the name of simplicity--just like how a thick winter outfit doesn't provide protection like padded armor.
So, here's what we know:
- Armor blocks Touch Attacks
- Clothing does not block Touch Attacks
- You can (apparently) use Touch Attacks through the 'glove' interior of a gauntlet.
Based on that, it's probably safe to assume that your character does not actually need to come into skin to skin contact in order to use Touch-range spells.
That leads to the last problem, which is the big one...what is "direct contact?" Well, grappling only requires one free hand, so you can 'grapple' by grabbing someone with a gauntleted hand. The fact that a grapple always causes a save against a Grung's poison means that their poison can apparently affect you straight through the glove of your gauntlet.
Based solely on this...anything that lets you communicate a Touch spell would also potentially transmit your poison. But the phrasing of "comes into direct contact with your skin" leaves the possibility of things like shoving a Grung into a burlap sack and not getting poisoned through the sack. So--if the Grung is wearing clothing, does it protect you from their skin? Is the special case with grappling because you have to grab and squeeze to hold them in place, so their poison is pressed through your glove?
All-in-all, the fact that it calls out a requirement to either grapple or come into direct contact with a Grung's skin leads me to believe that if a Grung wore clothing and was not handled roughly, the clothing would be able to contain their poison. This would allow you to physically contact party members in a way that would allow Touch-range spellcasting, without poisoning them. At the cost of preventing you from just touching enemies to poison them unless you took your gloves off first.
If you want to be absolutely certain/safe...then grab the Ritual Caster or Magic Initiate [Wizard] feats to get access to Find Familiar, and let your familiar do all the touch-range spellcasting for you.
This is far from a concrete ruling--you will absolutely need to ask your DM about this--but hopefully, this post gives you some discussion points you can bring up with them.