You might just want to leave them there and see how long they survive. Disturbing the azaleas might not be worth the trouble and risk of trying to untangle the bulbs.
If you give it a try, wait till about six weeks after the blossoms fade. Dig them out the best you can and replant where you want them. If at that point the floppy leaves look bad enough that they bother you, cut them back to an inch or two and let them continue to yellow and go dormant until next year.

If the previous owners planted the shrubs such a short time ago, and if the azaleas were pot-grown, then it is likely that the roots have barely begun to colonise the surrounding ground.

Pick a cool and somewhat damp day to lift the azaleas and place them in a shady place, sheltered from wind, with wet sacking over their roots while you dig up the bulbs.

You might want to rearrange the azaleas to get a more pleasing combination of colours so you could tag the plants before lifting them.

When they go back in the ground, remember to water well and mulch them for the coming summer months. They need to be firmed in, too, but not heavily. No Big Boots, please!

The other shrubs might not be so obliging if they have regular roots instead of fibrous root balls but it is worth checking. If the plants were pot-bound when they came from the plant store then it can take years (to never) for them to break out of that pot shape and start to really grow in your garden.

Thanks for the suggestions.

I was wondering if digging up the azaleas completely (to get at the bulbs) was an option.

If I just leave things alone, do you think the tulips are going to slow the azaleas growth?

No. The azaleas won't be affected by the bulbs. The tulips will fade away sooner or later because they won't be getting enough sun as the shrubs are growing and shading them.
Unless you really want the work, I wouldn't dig anything up unless it's some very rare bulb you need to save which is doubtful, lol.

Thanks, Sandy.

No, I don't particularly want to save the bulbs. I'm more interested in helping the azaleas.

Daffodils do naturalize, multiplying wherever they choose. I have some in the middle my azaleas too. It won't hurt the azalea. Sometimes nature knows best.