This is a full quote from Johnny's Seeds:

CULTURE:
Asparagus is a long-lived perennial vegetable crop. With proper site
selection, preparation, and care, it can thrive for 15 years or more.
Asparagus prefers well-drained, fertile soil with a pH of 7.0 in full
sun or part shade. Soil testing and bed preparation should be completed
in the season prior to planting. May not perform well in warmer areas
such as TX, MS, GA, LA, AL, or FL, as it may not become dormant, which
is needed to prevent frost/freeze damage to crowns in winter months.

GROWING FROM SEEDS:
Sow seeds indoors 12-14 weeks before desired transplant date. Sow 1
seed per 2" cell, ½-¾" deep. Germinate at 75-80°F (23-27°C) day, 65°F
(18°C) night. Fertilize moderately as needed. After last frost, plant
seedlings similarly to crowns, but use W-shaped furrow. Plant the
seedlings in the low middle point of the W to ensure better drainage.
Gradually fill in furrow as plants grow.

PLANTING CROWNS:
Crowns can be planted about 3-4 weeks prior to last average frost date.
In a furrow 5-8" deep (shallower for heavy soils), 12" wide, place
crowns with buds up and roots spread out, 8-14" apart in row (closer
spacing creates more slender spears), 3-6' between rows. Cover crowns
with 2-3" of soil at planting. As spears grow, gradually fill in furrow.
Use caution when cultivating to avoid damaging crowns. Keep bed free of
weeds (straw or leaf mulch works well) and irrigate regularly during
establishment. Allow "ferns" to grow – do not cut until they senesce in
the fall. Add compost or aged manure to the beds each fall or early
spring.

Bees, tall and falling over sounds like inadequate light. What kind of light do you have your seedings in?

They are getting inadequate light- or at least they are getting no artificial light. I lost my indoor grow setup space when I moved some family in, so I'm just using my south facing bay window. I've used this window in the past for other seedlings with no problems for late March starts- But I think with how early I started the asparagus and with the weather and day length not being up to snuff in Feb, that helped made a light difference that's inadequate.

But there's no help for that now, and they are tall. That's why I was wondering if I could transplant them deep like tomatoes or use stakes to stabilize them till I can get them outdoors.

I know this is an old thread, but I’m experiencing the same exact scenario. Did your seedlings survive?

Wow, I had forgotten about this thread.

So yes, a lot of them did survive. I ended up with around 36 of them in gallon pots. And of those I have around 20-24 of them still in their pots. Or at least did during cleanup last fall. They were supposed to be planted out already, but with some injuries and illness that planting got put on back burner.

I'm much recovered and will have some additional help setting them out this spring. Plan is to put them into the ground in an area out back. Not sure where, will have to see once the winter clears off and I can see what's already growing out there.

I can't remember for sure, I would have to really dig back through notes to see if I noted how I did it. But I probably did repot them from the TP tubes into solo cups for a while, then repotted again into the gallon pots.