Praise be to Allah.
Can you fast without ghusl?
The fast of one who has intercourse with his wife at night and wakes up in the morning in a state of janabah (major ritual impurity) is still valid, as is the fast of one who becomes junub in his sleep, whether at night or during the day.
There is nothing wrong with delaying ghusl until just before dawn. Rather fasting is invalidated by having intercourse during the day in Ramadan, from the onset of true dawn until the sun sets. (Fatawa al-Lajnah al-Daimah, 10/327)
Delaying prayer because of feeling shy to perform ghusl
With regard to your delaying the prayer until the sun had risen, this was wrong of you to do that. What is obligatory is to perform the prayers on time. Feeling too embarrassed to do ghusl is no excuse for delaying the prayer. You have to repent from that and pray for forgiveness.
May Allah help us and you to do all that is good.
And Allah knows best.
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