You should use a mortgage broker when applying for a mortgage in your situation. On base salary, you can get ~€350 but if you can show a pattern for overtime/commission then some lenders will include a percentage of it. With the right broker you will probably be able to borrow €400k without needing exemptions
Pay the loan off today. Then you know exactly where you stand with €10k in savings. And you really don't need to sell it for a cheaper car, that would be foolish.
I presume you both keep your finances separate and that is why one of you has the car loan?? But if you are serious about buying together, then you need to view your finances as one.
Are your expectations too high? A quick search of Daft shows plenty of properties (3 beds) below €400k mark.
As a first time buyer, you only need a 10% deposit. If you can get mortgage approval closer to €400k, then your target drops to €40k plus €10k for costs involved. Without cutting back too much, you should have that by mid/late next year. It would allow you to target property in the €370-420k range
The other big obvious help is both of your salaries and career progression. You need to be past probation in work to get a mortgage so if either of you are thinking of changing employers then try to do it sooner rather than later. Any increases or promotions will help with the amount you can borrow too so maybe there is some scope there, particularly for your partner
Overall you are in a far better place than most of your peers. You are still very young so don't overdo it on the savings to the point where you are saying no to everything. Enjoy life a little.