I know that pronouncing “щ” as “ш + ч” is outdated, but can I still pronounce it like that? Will people understand me?
My guess is that your ability to be understood wouldn't be greatly affected if you pronounced щ as [шч]. In fact, this pronunciation is a variant of the norm in words like счастье and счёт. Sometimes the listener might be confused as to whether you said щека or чека, for example.
If you're learning Russian, I suggest you try and learn the proper smooth pronunciation from day one. To my ear, the English sh is closer to щ than ш. If you think about it, щ is a palatalized and geminated (doubled) version of ш. The gemination is at least as important. It's probably best to practice saying minimal pairs like чаша - чаща, шит (sewn) - щит (shield). Keeping a straight face is part of the challenge!
In the rural middle Volga region, they tend to not pronounce щ, instead using composite consonants, but different ones.
Тысяча becomes [тышша], щи become [счи]. It's also a hard consonant: нищий is [нишшый].
(And no, they don't pronounce it quickly)
This doesn't cause any intelligibility issues, so the (West Slavic) [шч] should work, too. South Slavic [шт] much less so.
Nevertheless, if you don't make some other sounds just right, you'll be heading into "huh?" territory really fast.