A lot has already been written, now some facts. For analysis first, I do construction financing myself and work for a bank that only co-finances additional costs with management approval. So rather not.
There are many banks that finance additional costs (acquisition ancillary costs). There are also enough banks that "sneak in" private loans into the construction financing. This is not desired.
Existing loan installments generally do not interfere, IF the creditworthiness is sufficient.
From your net income, a self-retention (minimum retention) is deducted, this varies but is definitely at least 700 EUR for the first person and another 200 EUR for each additional person in the household. Operating costs are often calculated as a flat rate, realistically about 3 EUR /m². Then maintenance obligations, kindergarten, loans, etc. are accounted for. If there is still a positive household budget at the end, construction financing is generally possible.
By the way, the balance of the savings contracts is indeed considered equity capital. Also, building societies can grant smaller loans "blank," that is, only based on creditworthiness and not on collateral. This can improve the equity contribution at the bank. Also important here: Does the installment still fit? The building society loan must also be repaid, and this is quite high relative to a bank loan.