I'm going to approach the rod/wand issue from a RAW perspective.
Rules-wise, yes, you can use both. Except for the fact that you need a free hand to cast spells and you don't have a free hand. I suppose you could cast a spell that had no somatic or material component, but it turns out there are very few verbal-only spells that use an attack roll.
Ideally, the War Caster feat would solve that problem by allowing you to cast spells with your hands full. But it only allows you to cast while holding weapon or shield. Neither of those magic items is a weapon or a shield. So actually, it's of no help whatsoever.
The first big fix your DM might approve is to allow the Rod to double as a club. This is based on the 'improvised weapon' rules, especially the part about allowing a table leg, or any similar size item, to count as an actual club. Since it's now a weapon, the War Caster feat applies.
The second big fix is to allow either the Wand or the Rod to double as a Focus Item. That neatly allows you you cast somatic spells with hands full (but only if there's a material component), and doesn't even need a feat to make it work. Since wands and rods are both on the list of arcane foci, it's easy to argue.
If the DM allows both fixes, you can get away with some amazing things, for instance a combo like Shillelagh, Dissonant Whispers, and Booming Blade with War Caster to get some very nice opportunity attacks.
If you approach this from RAI, you very quickly run into one issue: The unwritten assumption seems to be that when you use the Wand, you are actually incorporating it into the somatic components. Specifically, you point with the wand and the spell shoots out through the wand, and that improves your aim. And possibly the same with the Rod, you tend to assume that it will be used in such a way that the spell is channelled through the rod. So it's hard to imagine using both at once.
In the end, almost every table house-rules the 'hands-free' and/or 'spell components' parts of the rules, because they really aren't written well enough to handle edge cases. So it will end up working as well or as badly as the DM wishes.