hi there --i'm actually in the exact situation you described in your post & am happy to discuss this with someone...
i started taking doxycycline (100mg 2x a day) a little over a month ago, but also started birth control pills (beyaz) about 2.5 weeks ago.
my ob/gyn told me that it's okay to finish off the remaining doxycycline that i have while simultaneously taking the birth control (since i'm taking it primarily for anti-acne purposes and not for actual birth control)...
like you, i've been SUPER worried about the breakout that you get after stopping doxycycline, so i was hoping that the birth control will pick up where the doxycycline left off and keep me clear. however, you must have also heard about the initial breakout you get when starting birth control (your hormones need to adjust during the first 3 months).
for me, the doxycycline helped my condition within the first week (i also started using two different topicals, benzamycin & azelex, since my derm said oral antibiotics are not entirely effective alone). i would say i had moderate acne all over my chin and sometimes big ones on my cheeks (which is also how i knew it was hormonal and immediately wanted to take birth control since i knew doxy wasn't going to be a permanent and healthy cure).
doxycycline + the topical prescriptions got me totally clear after 1 month... i was only left with scars and was sooo happy i had no bumps. i'm currently weaning myself off of doxy (started at 200mg a day, then 100mg a day, now 100mg every other day)... not entirely sure if this is effective or necessary at all, since i only read about this method online and my dermatologist never said anything about weaning off. she was just gonna let me stop cold turkey... maybe because whether you wean yourself off or stop cold turkey, the acne will come back either way, i believe. i don't want to scare you, but since i started lowering my doxy dose, i started to get large cysts in places that were not a problem before... on my jawline and high up on my cheekbone

. they're not like the acne i had before i started taking doxy (they're bigger and way more stubborn). and i've read about this same problem in other people's reviews... yea, it sucks that i can't stay clear during this weaning off process and i'm so worried about the scarring that i will have to deal with AGAIN after this whole doxy process is over. but i'm actually glad that i'm getting off of the doxy because it's just so inconvenient to take since you have to be so careful with what you eat/when/how much and i didn't like how it made me feel sometimes. and i'm also glad that i started the birth control while i'm weaning off the doxy because at least i know that it WILL get better eventually... i couldn't prevent the post-doxy breakout but at least i know the birth control will clear me up again eventually and it's also a healthier anti-acne option. my sister takes the same birth control as i do and she didn't have an initial breakout phase and also has been clear ever since she started a year ago... i'm hoping something similarly miraculous will happen to me.
so i know you already started the doxy (how is it going?)... and it probably will make you clear like it did for me, but like everyone says, it's only a temporary fix and it seems difficult to avoid the post-doxy breakout, even if you do take birth control afterwards (which in itself can cause its own initial breakout anyway). if i could go back, i would not have started the doxy at all and would have started with the birth control straight away, since it's known to be effective and i wouldn't have had to deal with the occasional nausea caused by doxy, etc. but either way, you'd most likely have to deal with some sort of breakout.