Hello, I am playing a hag-like character with a mix of 1st level artificer, 2nd level druid, and 2nd level wizard. Based on every calculation I've seen, my spell slots should go: 4 3. However, I am getting higher level slots than I should: 4 3 2. Additionally, despite having 2nd slots (or even the bugged 3rd level slots), it does not give me the option to prepare any druid or artificer spells above 1st level. Based on the player's handbook, "The spells must be of a level for which you have spell slots", and can change each long rest. Being fairly new to D&D Beyond, I can't seem to find a way to edit any of these things either.

Those are not bugged. That is the way Multiclassing works. An Artificer 1/Wiz 2/Druid 2 would count as a 5th level character when it comes to calculating how many spell slots you get, and of what levels. Therefore you are supposed to have 4 1st-level, 3 2nd-level, and 2 3rd-level spell slots.

However, when it comes to preparing spells, that still happens as if you were not a multiclassed character. Therefore you would prepare spells as if you were a 1st level Artificer. And you you would prepare spells as if you are a 2nd level Wizard. And you would prepare spells as if you were a 2nd level Druid. Since none of those can prepare spells of 2nd-level or higher, neither can you. You can use those 2nd and 3rd level spell slots to upcast any of the spells that you do have prepared however.

Spellcasting

Your capacity for spellcasting depends partly on your combined levels in all your spellcasting classes and partly on your individual levels in those classes. Once you have the Spellcasting feature from more than one class, use the rules below. If you multiclass but have the Spellcasting feature from only one class, you follow the rules as described in that class.

Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class. If you are a ranger 4/wizard 3, for example, you know three 1st-level ranger spells based on your levels in the ranger class. As 3rd-level wizard, you know three wizard cantrips, and your spellbook contains ten wizard spells, two of which (the two you gained when you reached 3rd level as a wizard) can be 2nd-level spells. If your Intelligence is 16, you can prepare six wizard spells from your spellbook.

Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.

If a cantrip of yours increases in power at higher levels, the increase is based on your character level, not your level in a particular class.

Spell Slots. You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature. Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.

If you have more than one spellcasting class, this table might give you spell slots of a level that is higher than the spells you know or can prepare. You can use those slots, but only to cast your lower-level spells. If a lower-level spell that you cast, like burning hands, has an enhanced effect when cast using a higher-level slot, you can use the enhanced effect, even though you don't have any spells of that higher level.

For example, if you are the aforementioned ranger 4/wizard 3, you count as a 5th-level character when determining your spell slots: you have four 1st-level slots, three 2nd-level slots, and two 3rd-level slots. However, you don't know any 3rd-level spells, nor do you know any 2nd-level ranger spells. You can use the spell slots of those levels to cast the spells you do know — and potentially enhance their effects.

Pact Magic. If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.

Multiclass Spellcaster

Level	1st	2nd	3rd	4th	5th	6th	7th	8th	9th
1st	2	-	-	-	-	-	-	-	-
2nd	3	-	-	-	-	-	-	-	-
3rd	4	2	-	-	-	-	-	-	-
4th	4	3	-	-	-	-	-	-	-
5th	4	3	2	-	-	-	-	-	-
6th	4	3	3	-	-	-	-	-	-
7th	4	3	3	1	-	-	-	-	-
8th	4	3	3	2	-	-	-	-	-
9th	4	3	3	3	1	-	-	-	-
10th	4	3	3	3	2	-	-	-	-
11th	4	3	3	3	2	1	-	-	-
12th	4	3	3	3	2	1	-	-	-
13th	4	3	3	3	2	1	1	-	-
14th	4	3	3	3	2	1	1	-	-
15th	4	3	3	3	2	1	1	1	-
16th	4	3	3	3	2	1	1	1	-
17th	4	3	3	3	2	1	1	1	1
18th	4	3	3	3	3	1	1	1	1
19th	4	3	3	3	3	2	1	1	1
20th	4	3	3	3	3	2	2	1	1

MULTICLASSING AND THE ARTIFICER

If your group uses the optional rule on multiclassing in the Player’s Handbook, here’s what you need to know if you choose artificer as one of your classes.

Ability Score Minimum. As a multiclass character, you must have at least an Intelligence score of 13 to take a level in this class, or to take a level in another class if you are already an artificer.

Proficiencies Gained. If artificer isn’t your initial class, here are the proficiencies you gain when you take your first level as an artificer: light armor, medium armor, shields, thieves’ tools, tinker’s tools.

Spell Slots. Add half your levels (rounded up) in the artificer class to the appropriate levels from other classes to determine your available spell slots.

Thank you. I didn't notice that Artificer's round up and that the rules for preparing multiclass spells were explained different than what it says on the class page. 

Yeah, the other “half casters” (Ranger and Paladin) don’t get Spellcasting until 2nd level, but Artificers get it at 1st level so they have to round up while the others round down.